ALGEBRA Quadrilateral ABCD is a rhombus.
Find each value or measure.
1.If , find .
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So, Then, isan
isosceles triangle. Therefore,
If a parallelogram is a rhombus, then each diagonal
bisects a pair of opposite angles. So,
.
Therefore,
ANSWER:
32
2.If AB = 2x + 3 and BC = x + 7, find CD.
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So,
So, AB = 2(4) + 3 = 11.
CD is congruent to AB, so CD = 11.
ANSWER:
11
3.PROOF Write a two-column proof to prove that if
ABCD is a rhombus with diagonal
.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rhombus with
diagonal . You need to prove . Use the
properties that you have learned about rhombi to
walk through the proof.
Given: ABCD is a rhombus with diagonal .
Prove:
Proof:
Statements(Reasons)
1. ABCD is a rhombus with diagonal . (Given)
2. (Diag.ofrhombusbisects )
3. (Refl.Prop.)
4. (Def.ofrhombus)
5. (SAS)
6. (CPCTC)
ANSWER:
Given: ABCD is a rhombus with diagonal .
Prove:
Proof:
Statements(Reasons)
1. ABCD is a rhombus with diagonal . (Given)
2. (Diag.ofrhombusbisects )
3. (Refl.Prop.)
4. (Def.ofrhombus)
5. (SAS)
6. (CPCTC)
4.GAMES Thecheckerboardbelowismadeupof64
congruent black and red squares. Use this
information to prove that the board itself is a square.
SOLUTION:
Sampleanswer:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given that the checkerboard is
made up of 64 congruent squares. You need to prove
that the board is a square. Use the properties that
you have learned about squares to walk through the
proof.
Since each side of the board is 8 squares in length
and each of the squares is congruent, the lengths of
all four sides of the board are equal. Since we know
that each of the four quadrilaterals that form the
corners of the board are squares, we know that the
measure of the angle of each vertex of the board is
90. Therefore, the board is a square.
ANSWER:
Sample answer: Since each side of the board is 8
squares in length and each of the squares is
congruent, the lengths of all four sides of the board
are equal. Since we know that each of the four
quadrilaterals that form the corners of the board are
squares, we know that the measure of the angle of
each vertex of the board is 90. Therefore, the board
is a square.
COORDINATE GEOMETRY Given each set
ofvertices,determinewhether QRST is a
rhombus, a rectangle, or a square. List all that
apply. Explain.
5.Q(1, 2), R(–2, –1), S(1, –4), T(4, –1)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
So, the parallelogram is a rectangle. Check whether
thetwodiagonalsareperpendicular.
QS has a slope of .
RT has a slope of . These slopes are opposite
reciprocals.
The diagonals are perpendicular. So, it is a rhombus.
Since the diagonals are both congruent and
perpendicular to each other the parallelogram is a
rectangle, rhombus and square.
ANSWER:
Rectangle, rhombus, square; consecutive sides are
, all sides are .
6.Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are not perpendicular. So, it is not a
rhombus either.
ANSWER:
None; the diagonals are not congruent or
perpendicular.
ALGEBRA Quadrilateral ABCD is a rhombus.
Find each value or measure.
7.If AB = 14, find BC.
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So,
Therefore, BC = 14.
ANSWER:
14
8.If , find .
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So, Then, isan
isosceles triangle. Therefore,
If a parallelogram is a rhombus, then each diagonal
bisects a pair of opposite angles. So,
Therefore,
ANSWER:
27
9.If AP = 3x – 1 and PC = x + 9, find AC.
SOLUTION:
Thediagonalsofarhombusbisecteachother.
3x – 1 = x + 9
2x = 10
x = 5
Therefore, AC = 2(5 + 9) = 28.
ANSWER:
28
10.If DB = 2x – 4 and PB = 2x – 9, find PD.
SOLUTION:
The diagonals of a rhombus bisect each other. So,
Therefore, PD = PB = 2(7) – 9 = 5.
ANSWER:
5
11.If , find
.
SOLUTION:
In a rhombus, consecutive interior angles are
supplementary.
Each pair of opposite angles of a rhombus is
congruent. So,
ANSWER:
95
12.If , find x.
SOLUTION:
The diagonals of a rhombus are perpendicular to
each other.
ANSWER:
35
CCSSARGUMENTSWriteatwo-column
proof.
13.Given:
Prove: WXYZ is a rhombus.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are
given . You
need to prove that WXYZ is a rhombus. Use the
properties that you have learned about parallelograms
and rhombi to walk through the proof.
Given:
Prove: WXYZ is a rhombus.
Proof:
Statements(Reasons)
1. (Given)
2. WXYZ is a . (Both pairs of opp. sides are .)
3. WXYZ is a rhombus. (If one pair of consecutive
sides of a are , the isarhombus.)
ANSWER:
Given:
Prove: WXYZ is a rhombus.
Proof:
Statements(Reasons)
1. (Given)
2. WXYZ is a . (Both pairs of opp. sides are .)
3. WXYZ is a rhombus. (If one pair of consecutive
sides of a are , the isarhombus.)
14.Given: QRST is a parallelogram.
Prove: QRST is a square.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove.Here,youaregivenQRST is a parallelogram;
. You need to prove that
QRST is a square. Use the properties that you have
learned about parallelograms and squares to walk
through the proof.
Given: QRST is a parallelogram;
Prove: QRST is a square.
Proof:
Statements(Reasons)
1. QRST is a parallelogram;
. (Given)
2. QRST is a rectangle. (If the diagonals of a
, the isarectangle.)
3. isarightangle.(Defofrt. )
4. (Def.ofperpendicular)
5. QRST is a rhombus. (If the diagonals of a
isarhombus.)
6. QRST is a square. (Thm.6.2, if a quadrilateral is a
rectangle and a rhombus, then it is a square.)
ANSWER:
Given: QRST is a parallelogram;
Prove: QRST is a square.
Proof:
Statements(Reasons)
1. QRST is a parallelogram; . (Given)
2. QRST is a rectangle. (If the diagonals of a
, the isarectangle.)
3. isarightangle.(Defofrt. )
4. (Def.ofperpendicular)
5. QRST is a rhombus. (If the diagonals of a
, isarhombus.)
6. QRST is a square. (Thm.6.2, if a quadrilateral is a
rectangle and a rhombus, then it is a square.)
15.Given: JKQP is a square. .
Prove: JKLM is a parallelogram.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove.Here,youaregiven
JKQP is a square and . You
need to prove that JKLM is a parallelogram. Use the
properties that you have learned about parallelograms
to walk through the proof.
Given: JKQP is a square.
.
Prove: JKLM is a parallelogram.
Proof:
Statements (Reasons)
1. JKQP is a square. .
(Given)
2. JKQP is a parallelogram. (All squares are
parallelograms.)
3. (Def.of )
4. (Opp.Sidesof .)
5. JP = KQ (Def of segs.)
6. JM = MP, KL = LQ (Def. of bisects)
7. JP = JM + MP, KQ = KL + LQ (Seg. Add Post.)
8. JP = 2JM, KQ = 2KL (Subst.)
9. 2JM = 2KL (Subst.)
10. JM = KL (Division Prop.)
11. (Def.of segs.)
12. JKLM is a parallelogram. (If one pair of opp.
sides is , then the quad. is a .)
ANSWER:
Proof:
Statements (Reasons)
1. JKQP is a square. .
(Given)
2. JKQP is a parallelogram. (All squares are
parallelograms.)
3. (Def.of )
4. (Opp.sidesofof .)
5. JP = KQ (Def of segs.)
6. JM = MP, KL = LQ (Def. of bisects)
7. JP = JM + MP, KQ = KL + LQ (Seg. Add Post.)
8. JP = 2JM, KQ = 2KL (Subst.)
9. 2JM = 2KL (Subst.)
10. JM = KL (Division Prop.)
11. (Def.of segs.)
12. JKLM is a parallelogram. (If one pair of opp.
sides is , then the quad. is a .)
16.Given: ACDH and BCDF are parallelograms;
.
Prove: ABFH is a rhombus.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ACDH and BCDF are
parallelograms; .You need to prove that
ABHF is a rhombus. Use the properties that you
have learned about parallelograms and rhombi to
walk through the proof.
Given: ACDH and BCDF are parallelograms;
.
Prove: ABHF is a rhombus.
Proof:
Statements (Reasons)
1. ACDH and BCDF are parallelograms; .
(Given)
2. (Def.of )
3. (Trans.Prop)
4. (Def.of )
5. AC = HD (Def of segs.)
6. AC = AB + BC, HD = HF + FD (Seg. Add. Post.)
7. AC – HD = AB + BC – HF – FD (Subt. Prop.)
8. AB = HF (Subst.)
9. (Def.of segs.)
10. (Subst.)
11. ABFH is a rhombus. (Def. of rhombus)
ANSWER:
Proof:
Statements (Reasons)
1. ACDH and BCDF are parallelograms; .
(Given)
2. (Def.of )
3. (Trans.Prop)
4. (Def.of )
5. AC = HD (Def of segs.)
6. AC = AB + BC, HD = HF + FD (Seg. Add. Post.)
7. AC – HD = AB + BC – HF – FD (Subt. Prop.)
8. AB = HF (Subst.)
9. (Def.of segs.)
10. (Subst.)
11. ABFH is a rhombus. (Def. of rhombus)
17.ROADWAYS Main Street and High Street intersect
as shown in the diagram. Each of the crosswalks is
the same length. Classify the quadrilateral formed by
the crosswalks. Explain your reasoning.
SOLUTION:
The two streets intersect at a 29 degree angle.
Analyze this first to determine the measure of an
angle of the quadrilateral. Then analyze the
information given about the lengths of the crosswalks
to categorize the quadrilateral formed.
The measure of the angle formed between the two
streets is 29, and vertical angles are congruent, so
the measure of one angle of the quadrilateral is 29.
So the quadrilateral is not a rectangle or square.
Since the crosswalks are the same length, the sides
of the quadrilateral are congruent. Therefore, they
form a rhombus.
ANSWER:
Rhombus; Sample answer: The measure of angle
formed between the two streets is 29, and vertical
angles are congruent, so the measure of one angle of
the quadrilateral is 29. Since the crosswalks are the
same length, the sides of the quadrilateral are
congruent. Therefore, they form a rhombus.
18.CCSS MODELING A landscaper has staked out
the area for a square garden as shown. She has
confirmed that each side of the quadrilateral formed
by the stakes is congruent and that the diagonals are
perpendicular. Is this information enough for the
landscaper to be sure that the garden is a square?
Explain your reasoning.
SOLUTION:
Compare the information given to the criteria to
classify a quadrilateral as a square.
Since the four sides of the quadrilateral are
congruent and the diagonals are perpendicular, the
figure is either a square or a rhombus. To be sure
that the garden is a square, she also needs to confirm
that the diagonals are congruent.
ANSWER:
No; sample answer: Since the four sides of the
quadrilateral are congruent and the diagonals are
perpendicular, the figure is either a square or a
rhombus. To be sure that the garden is a square, she
also needs to confirm that the diagonals are
congruent.
COORDINATE GEOMETRY Given each set
of vertices, determine whether JKLM is a
rhombus, a rectangle, or a square. List all that
apply. Explain.
19.J(–4, –1), K(1, –1), L(4, 3), M(–1, 3)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are perpendicular. So, it is a rhombus.
ANSWER:
Rhombus; the diagonals are .
20.J(–3, –2), K(2, –2), L(5, 2), M(0, 2)
SOLUTION:
First, graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are perpendicular. So, it is a rhombus.
ANSWER:
Rhombus; the diagonals are .
21.J(–2, –1), K(–4, 3), L(1, 5), M(3, 1)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are not perpendicular. So, it is not a
rhombus either.
ANSWER:
None; the diagonals are not .
22.J(–1, 1), K(4, 1), L(4, 6), M(–1, 6)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are congruent. So, the parallelogram is
a rectangle. Check whether the two diagonals are
perpendicular.
The diagonals are perpendicular. So, it is a rhombus.
Since the diagonals are both congruent and
perpendicular to each other the parallelogram is a
rectangle, rhombus and square.
ANSWER:
Square, rectangle, rhombus; all sides are .
ABCD is a rhombus. If PB = 12, AB = 15, and
, find each measure.
23.AP
SOLUTION:
The diagonals of a rhombus are perpendicular to
each other. So, by the Pythagorean Theorem, AP2 =
AB2 – PB2.
ANSWER:
9
24.CP
SOLUTION:
All the four sides of a rhombus are congruent and the
diagonalsareperpendiculartoeachother.
So, by the Pythagorean Theorem, CP2 = BC2 – PB2.
BC = AB. Substitute AB for BC.
ANSWER:
9
25.
SOLUTION:
All the four sides of a rhombus are congruent. So,
isanisoscelestriangle.Then,
ANSWER:
24
26.
SOLUTION:
The diagonals are perpendicular to each other. So, in
the right triangle PAB,
All the four sides of a rhombus are congruent. So,
isanisoscelestriangle.Then,
ANSWER:
66
WXYZ is a square. If WT = 3, find each
measure.
27.ZX
SOLUTION:
The diagonals of a square are congruent and bisect
each other.
So, ZX = WY = 2(WT) = 6.
ANSWER:
6
28.XY
SOLUTION:
The diagonals of a square are congruent and bisect
each other at right angles.
So, YT = XT = WT = 3.
By the Pythagorean Theorem, XY2 = YT2 + XT2.
ANSWER:
29.
SOLUTION:
The diagonals of a square are perpendicular to each
other.
So,
ANSWER:
90
30.
SOLUTION:
In a square, each diagonal bisects a pair of opposite
angles. So,
ANSWER:
45
Classify each quadrilateral.
31.Refer to the photo on p. 432.
SOLUTION:
The diagonals are congruent and bisect each other at
right angle. Therefore, the quadrilateral is a square.
ANSWER:
square
32.Refer to the photo on p. 432.
SOLUTION:
The two pairs of opposite angles are congruent and
the adjacent sides are congruent. Therefore, the
quadrilateral is a rhombus.
ANSWER:
rhombus
33.Refer to the photo on p. 432.
SOLUTION:
The two pairs of opposite sides are congruent and
one of the angles is a right angle. Therefore, the
quadrilateral is a rectangle.
ANSWER:
rectangle
PROOF Write a paragraph proof.
34.Theorem 6.16
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rhombus. You
need to prove that each diagonal bisects a pair of
opposite angles.. Use the properties that you have
learned about rhombi to walk through the proof.
Given: ABCD is a rhombus.
Prove: Each diagonal bisects a pair of opposite
angles.
Proof: We are given that ABCD is a rhombus. By
definition of rhombus, ABCD is a parallelogram.
Opposite angles of a parallelogram are congruent, so
.
becauseallsidesofarhombus
are congruent. bySAS.
byCPCTC.
bySAS. by
CPCTC. By definition of angle bisector, each
diagonal bisects a pair of opposite angles.
ANSWER:
Given: ABCD is a rhombus.
Prove: Each diagonal bisects a pair of opposite
angles.
Proof: We are given that ABCD is a rhombus. By
definition of rhombus, ABCD is a parallelogram.
Opposite angles of a parallelogram are congruent, so
.
becauseallsidesofarhombus
are congruent. bySAS.
byCPCTC.
bySAS. by
CPCTC. By definition of angle bisector, each
diagonal bisects a pair of opposite angles.
35.Theorem 6.17
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a parallelogram;
. You need to prove that ABCD is a
rhombus. Use the properties that you have learned
about rhombi to walk through the proof.
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: We are given that ABCD is a parallelogram.
The diagonals of a parallelogram bisect each other,
so becausecongruenceof
segments is reflexive. We are also given that
. Thus, areright
angles by the definition of perpendicular lines. Then
becauseallrightanglesare
congruent. Therefore, bySAS.
byCPCTC.Oppositesidesof
parallelograms are congruent, so
. Then since congruence of
segments is transitive, . All
four sides of ABCD are congruent, so ABCD is a
rhombus by definition.
ANSWER:
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: We are given that ABCD is a parallelogram.
The diagonals of a parallelogram bisect each other,
so becausecongruenceof
segments is reflexive. We are also given that
. Thus, areright
angles by the definition of perpendicular lines. Then
becauseallrightanglesare
congruent. Therefore, bySAS.
byCPCTC.Oppositesidesof
parallelograms are congruent, so
. Then since congruence of
segments is transitive, . All
four sides of ABCD are congruent, so ABCD is a
rhombus by definition.
36.Theorem 6.18
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a diagonal of a
parallelogram bisects an angle of a parallelogram.
You need to prove that the parallelogram is a
rhombus. Use the properties that you have learned
about parallelograms and rhombi to walk through the
proof.
Given: ABCD is a parallelogram; diagonal
bisects .
Prove: ABCD is a rhombus.
Proof: It is given that ABCD is a parallelogram. Since
opposite sides of a parallelogram are parallel,
. By definition, arealternate
interior angles of parallel sides . Since
alternate interior angles are congruent, .
Congruence of angles is symmetric, therefore
. it is given that bisects
, so by
definition. By the Transitive Property,
.
The sides opposite congruent angles in a triangle are
congruent, therefore, . So,
since a pair of consecutive sides of the parallelogram
is congruent, ABCD is a rhombus.
ANSWER:
If a diagonal of a parallelogram bisects an angle of a
parallelogram, then the parallelogram is a rhombus.
Given: ABCD is a parallelogram; diagonal
bisects .
Prove: ABCD is a rhombus.
Proof: It is given that ABCD is a parallelogram. Since
opposite sides of a parallelogram are parallel,
. By definition, arealternate
interior angles of parallel sides . Since
alternate interior angles are congruent, .
Congruence of angles is symmetric, therefore
. it is given that bisects
, so by
definition. By the Transitive Property,
.
The sides opposite congruent angles in a triangle are
congruent, therefore, . So,
since a pair of consecutive sides of the parallelogram
is congruent, ABCD is a rhombus.
37.Theorem 6.19
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a parallelogram;
. You need to prove that ABCD is a
rhombus. Use the properties that you have learned
about rhombi to walk through the proof.
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: Opposite sides of a parallelogram are
congruent, so . We are
given that . So, by the Transitive Property,
. So, . Thus, ABCD
is a rhombus by definition.
ANSWER:
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: Opposite sides of a parallelogram are
congruent, so . We are
given that . So, by the Transitive Property,
. So, . Thus, ABCD
is a rhombus by definition.
38.Theorem 6.20
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rectangle and
a rhombus. You need to prove that ABCD is a
square. Use the properties that you have learned
about squares to walk through the proof.
Given: ABCD is a rectangle and a rhombus.
Prove: ABCD is a square.
Proof: We know that ABCD is a rectangle and a
rhombus. ABCD is a parallelogram, since all
rectangles and rhombi are parallelograms. By the
definition of a rectangle, are
right angles. By the definition of a rhombus, all of the
sides are congruent. Therefore, ABCD is a square
since ABCD is a parallelogram with all four sides
congruent and all the angles are right.
ANSWER:
Given: ABCD is a rectangle and a rhombus.
Prove: ABCD is a square.
Proof: We know that ABCD is a rectangle and a
rhombus. ABCD is a parallelogram, since all
rectangles and rhombi are parallelograms. By the
definition of a rectangle, are
right angles. By the definition of a rhombus, all of the
sides are congruent. Therefore, ABCD is a square
since ABCD is a parallelogram with all four sides
congruent and all the angles are right.
CONSTRUCTION Use diagonals to construct
each figure. Justify each construction.
39.rhombus
SOLUTION:
Sample answer: Construct the perpendicular bisector
of a line segment. Place the compass at the midpoint
of the segment. Use the same compass setting to
locate a point on the perpendicular bisector above
and below the segment. Connect the endpoints of the
segments with the two points on the perpendicular
bisectortoformaquadrilateral.
Since the diagonals bisect each other, the
quadrilateral is a parallelogram. If the diagonals of a
parallelogram are perpendicular to each other, then
the parallelogram is a rhombus. Thus, the constructed
quadrilateral is a rhombus.
ANSWER:
Sample answer: The diagonals bisect each other, so
the quadrilateral is a parallelogram. Since the
diagonals of the parallelogram are perpendicular to
each other, the parallelogram is a rhombus.
40.square
SOLUTION:
Sample answer: Construct the perpendicular bisector
of a segment. Adjust the compass setting to equal the
distance from the midpoint of the segment to one of
its endpoints. Place the compass at the midpoint of
the segment. Draw arcs that intersect the
perpendicular bisector above and below the segment.
Connect the two points of intersection with the
endpoints of the segment to form a quadrilateral.
Since the diagonals bisect each other, the
quadrilateral is a parallelogram. If the diagonals of a
parallelogram are congruent and perpendicular, then
the parallelogram is a square. Thus, the constructed
quadrilateral is a square.
ANSWER:
Sample answer: The diagonals bisect each other, so
the quadrilateral is a parallelogram. Since the
diagonals of the parallelogram are congruent and
perpendicular, the parallelogram is a square.
PROOF Write a coordinate proof of each
statement.
41.The diagonals of a square are perpendicular.
SOLUTION:
Begin by positioning square ABCD on a coordinate
plane. Place vertex A at the origin. Let the length of
the bases be a units. Then the rest of the vertices are
B(a, 0), C(a, a), and D(0, a). You need to walk
through the proof step by step. Look over what you
are given and what you need to prove. Here, you are
given ABCD is a square and you need to prove that
. Use the properties that you have learned
about squares to walk through the proof.
Given: ABCD is a square.
Prove:
Proof:
slope of
slope of
The slope of is the negative reciprocal of the
slope of , so they are perpendicular.
ANSWER:
Given: ABCD is a square.
Prove:
Proof:
slope of
slope of
The slope of is the negative reciprocal of the
slope of , so they are perpendicular.
42.The segments joining the midpoints of the sides of a
rectangle form a rhombus.
SOLUTION:
Begin by positioning rectangle ABCD on a coordinate
plane. Place vertex A at the origin. Let the length of
the bases be a units and the height be b units. Then
the rest of the vertices are B(a, 0), C(a, b), and D(0,
b). You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rectangle and
Q, R, S, and T are midpoints of their respective
sides.and you need to prove that QRST is a rhombus.
Use the properties that you have learned about
rhombi to walk through the proof.
Given: ABCD is a rectangle. Q, R, S, and T are
midpoints of their respective sides.
Prove: QRST is a rhombus.
Proof:
Midpoint Q is .
Midpoint R is .
Midpoint S is .
Midpoint T is .
QR = RS = ST = QT
QRST is a rhombus.
ANSWER:
Given: ABCD is a rectangle. Q, R, S, and T are
midpoints of their respective sides.
Prove: QRST is a rhombus.
Proof:
Midpoint Q is .
Midpoint R is .
Midpoint S is .
Midpoint T is .
QR = RS = ST = QT
QRST is a rhombus.
43.DESIGN The tile pattern below consists of regular
octagons and quadrilaterals. Classify the
quadrilaterals in the pattern and explain your
reasoning.
SOLUTION:
In order to classify the quadrilaterals we need
information about the interior angles and the sides.
It's given that each quadrilateral is formed by 4
regular octagons. We can use what we know about
the exterior angles of a regular octagon as well as
the sides of a regular octagon to determine which
type of quadrilateral is in the pattern.
Squares; sample answer: Since the octagons are
regular each side is congruent, and the quadrilaterals
share common sides with the octagon, so the
quadrilaterals are either rhombi or squares. The
vertices of the quadrilaterals are formed by the
exterior angles of the sides of the octagons adjacent
to the vertices. The sum of the measures of the
exterior angles of a polygon is always 360 and since
a regular octagon has 8 congruent exterior angles,
each one measures 45. As shown in the diagram,
each angle of the quadrilaterals in the pattern
measures 45 + 45 or 90. Therefore, the quadrilateral
is a square.
ANSWER:
Squares; sample answer: Since the octagons are
regular each side is congruent, and the quadrilaterals
share common sides with the octagon, so the
quadrilaterals are either rhombi or squares. The
vertices of the quadrilaterals are formed by the
exterior angles of the sides of the octagons adjacent
to the vertices. The sum of the measures of the
exterior angles of a polygon is always 360 and since
a regular octagon has 8 congruent exterior angles,
each one measures 45. As shown in the diagram,
each angle of the quadrilaterals in the pattern
measures 45 + 45 or 90. Therefore, the quadrilateral
is a square.
44.REPAIR The window pane shown needs to be
replaced. What are the dimensions of the
replacement pane?
SOLUTION:
The window pane is in the shape of a square. The
diagonal of a square is the hypotenuse of a right
triangle with two consecutive sides of the square as
its legs. Let x be the length of each side of the
square. So, by the Pythagorean Theorem,
Therefore, the length of each side of the square is
about 15 inches.
ANSWER:
square; 15 in.
45.MULTIPLE REPRESENTATIONS In this
problem, you will explore the properties of kites,
which are quadrilaterals with exactly two distinct
pairs of adjacent congruent sides.
a. GEOMETRIC Draw three kites with varying
side lengths. Label one kite ABCD, one PQRS, and
one WXYZ. Then draw the diagonals of each kite,
labeling the point of intersection N for each kite.
b. TABULAR Measure the distance from N to
each vertex. Record your results in a table like the
one shown.
c. VERBAL Make a conjecture about the diagonals
of a kite.
SOLUTION:
a. Sample answer: Draw three different kites, each
with the intersection of the diagonals labeled N.
b.Use a ruler to measure each length in the table.
c. Sample answer: From the measurements recorded
in the table, N is the midpoint of each of the short
diagonals. The shorter diagonal of a kite is bisected
by the longer diagonal.
ANSWER:
a. Sample answer:
b.
c. Sample answer: The shorter diagonal of a kite is
bisected by the longer diagonal.
46.ERROR ANALYSIS In quadrilateral PQRS,
. Lola thinks that the quadrilateral is a
square, and Xavier thinks that it is a rhombus. Is
either of them correct? Explain your reasoning.
SOLUTION:
The only information known is that the diagonals are
congruent. Review the quadrilaterals that have
congruent diagonals. Is this enough information to
classify the quadrilateral?
Since they do not know that the sides of the
quadrilateral are congruent, only that the diagonals
are congruent, they can only conclude that the
quadrilateral is a rectangle. So, neither of them are
correct.
ANSWER:
Neither; sample answer: Since they do not know that
the sides of the quadrilateral are congruent, only that
the diagonals are congruent, they can only conclude
that the quadrilateral is a rectangle.
47.CCSS ARGUMENTS Determine whether the
statement is true or false. Then write the converse,
inverse, and contrapositive of the statement and
determine the truth value of each. Explain your
reasoning.
If a quadrilateral is a square, then it is a
rectangle.
SOLUTION:
True; sample answer: A rectangle is a quadrilateral
with four right angles and a square is both a
rectangle and a rhombus, so a square is always a
rectangle.
Converse: If a quadrilateral is a rectangle then it is a
square. False; sample answer: A rectangle is a
quadrilateral with four right angles. It is not
necessarily a rhombus, so it is not necessarily a
square.
Inverse: If a quadrilateral is not a square, then it is
not a rectangle. False; sample answer: A
quadrilateral that has four right angles and two pairs
of congruent sides is not a square, but it is a
rectangle.
Contrapositive: If a quadrilateral is not a rectangle,
then it is not a square. True; sample answer: If a
quadrilateral is not a rectangle, it is also not a square
by definition.
ANSWER:
True; sample answer: A rectangle is a quadrilateral
with four right angles and a square is both a
rectangle and a rhombus, so a square is always a
rectangle.
Converse: If a quadrilateral is a rectangle then it is a
square. False; sample answer: A rectangle is a
quadrilateral with four right angles. It is not
necessarily a rhombus, so it is not necessarily a
square.
Inverse: If a quadrilateral is not a square, then it is
not a rectangle. False; sample answer: A
quadrilateral that has four right angles and two pairs
of congruent sides is not a square, but it is a
rectangle.
Contra positive: If a quadrilateral is not a rectangle,
then it is not a square. True; sample answer: If a
quadrilateral is not a rectangle, it is also not a square
by definition.
48.CHALLENGE The area of square ABCD is 36
square units and the area of is 20 square
units.If , find the length of
.
SOLUTION:
Since the area of the square is 36 square units, the
length of each side of the square is 6 units. All the
four angles of a square are right angles. So, by the
Pythagorean Theorem,
The area of is20squareunits.So,
Also, we have
So, by HL postulate,
AE = CF by CPCTC
Therefore, CF = 2.
ANSWER:
2
49.OPEN ENDED Find the vertices of a square with
diagonals that are contained in the lines y = x and y =
−x + 6. Justify your reasoning.
SOLUTION:
Sampleanswer:
First graph the lines y = x and y = -x + 6. There are 6
units from the origin to the y-intercept of y = -x + 6
and 6 units from the origin to the x-intercept of y = -x
+ 6. So, three of the vertices of a square will be at (0,
0), ((0, 6), and (6, 0). The point 6 units above (6, 0) is
(6, 6).
(0, 0), (6, 0), (0, 6), (6, 6); the diagonals are
perpendicular, and any four points on the lines
equidistant from the intersection of the lines will be
the vertices of a square.
ANSWER:
Sample answer: (0, 0), (6, 0), (0, 6), (6, 6); the
diagonals are perpendicular, and any four points on
the lines equidistant from the intersection of the lines
will be the vertices of a square.
50.WRITING IN MATH Compare all of the propertie
rectangles, rhombi, and squares.
SOLUTION:
Sample answer:
Parallelogram: Opposite sides of a parallelogram are
parallelogram are congruent. The diagonals of a paral
separates a parallelogram into two congruent triangle
Rectangle: A rectangle has all the properties of a par
The diagonals of a rectangle are congruent.
Rhombus: A rhombus has all of the properties of a pa
congruent. The diagonals of a rhombus are perpendic
Square: A square has all of the properties of a paralle
rectangle. A square has all of the properties of a rho
ANSWER:
Sample answer:
Parallelogram: Opposite sides of a parallelogram are
parallelogram are
congruent. The diagonals of a parallelogram bisect ea
parallelogram into two congruent triangles.
Rectangle: A rectangle has all the properties of a par
The diagonals of a rectangle are congruent.
Rhombus: A rhombus has all of the properties of a pa
congruent. The diagonals of a rhombus are perpendic
Square: A square has all of the properties of a paralle
rectangle. A square has all of the properties of a rho
51.JKLM is a rhombus. If CK = 8 and JK = 10, find JC.
A 4C 8
B 6D 10
SOLUTION:
The diagonals of a rhombus are perpendicular to
each other. So, by the Pythagorean Theorem,
JC2 = JK2 – CK2.
Therefore, the correct choice is B.
ANSWER:
B
52.EXTENDED RESPONSE The sides of square
ABCD are extended by sides of equal length to form
square WXYZ.
a. If CY = 3 cm and the area of ABCD is 81 cm2,
find the area of WXYZ.
b. If the areas of ABCD and WXYZ are
respectively, find DZ.
c. If AB = 2CY and the area of ABCD = g square
meters, find the area of WXYZ in square meters.
SOLUTION:
a. Since the area of the square is 81 square cm, the
length of each side of the square is 9 cm. Since the
sides of the square ABCD are extended to form
square WXYZ, DZ = AW = BX = CY = 3 cm. Area of
each of the triangle is
The area of the square WXYZ is the sum of the 4
congruent triangles and the area of the square.
b. In the right triangle WZD, WZ = 13 and WD = AW
+ AD = DZ + 7 = x + 7 where x is the length of the
segment BythePythagoreanTheorem,
Solve the equation for x.
Since x is a length, it cannot be negative. Therefore,
the length DZ is about 5 cm.
c. If AB = 2CY, the lengths of the two legs of the
are
are
,
are
are
are
ALGEBRA Quadrilateral ABCD is a rhombus.
Find each value or measure.
1.If , find .
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So, Then, isan
isosceles triangle. Therefore,
If a parallelogram is a rhombus, then each diagonal
bisects a pair of opposite angles. So,
.
Therefore,
ANSWER:
32
2.If AB = 2x + 3 and BC = x + 7, find CD.
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So,
So, AB = 2(4) + 3 = 11.
CD is congruent to AB, so CD = 11.
ANSWER:
11
3.PROOF Write a two-column proof to prove that if
ABCD is a rhombus with diagonal
.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rhombus with
diagonal . You need to prove . Use the
properties that you have learned about rhombi to
walk through the proof.
Given: ABCD is a rhombus with diagonal .
Prove:
Proof:
Statements(Reasons)
1. ABCD is a rhombus with diagonal . (Given)
2. (Diag.ofrhombusbisects )
3. (Refl.Prop.)
4. (Def.ofrhombus)
5. (SAS)
6. (CPCTC)
ANSWER:
Given: ABCD is a rhombus with diagonal .
Prove:
Proof:
Statements(Reasons)
1. ABCD is a rhombus with diagonal . (Given)
2. (Diag.ofrhombusbisects )
3. (Refl.Prop.)
4. (Def.ofrhombus)
5. (SAS)
6. (CPCTC)
4.GAMES Thecheckerboardbelowismadeupof64
congruent black and red squares. Use this
information to prove that the board itself is a square.
SOLUTION:
Sampleanswer:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given that the checkerboard is
made up of 64 congruent squares. You need to prove
that the board is a square. Use the properties that
you have learned about squares to walk through the
proof.
Since each side of the board is 8 squares in length
and each of the squares is congruent, the lengths of
all four sides of the board are equal. Since we know
that each of the four quadrilaterals that form the
corners of the board are squares, we know that the
measure of the angle of each vertex of the board is
90. Therefore, the board is a square.
ANSWER:
Sample answer: Since each side of the board is 8
squares in length and each of the squares is
congruent, the lengths of all four sides of the board
are equal. Since we know that each of the four
quadrilaterals that form the corners of the board are
squares, we know that the measure of the angle of
each vertex of the board is 90. Therefore, the board
is a square.
COORDINATE GEOMETRY Given each set
ofvertices,determinewhether QRST is a
rhombus, a rectangle, or a square. List all that
apply. Explain.
5.Q(1, 2), R(–2, –1), S(1, –4), T(4, –1)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
So, the parallelogram is a rectangle. Check whether
thetwodiagonalsareperpendicular.
QS has a slope of .
RT has a slope of . These slopes are opposite
reciprocals.
The diagonals are perpendicular. So, it is a rhombus.
Since the diagonals are both congruent and
perpendicular to each other the parallelogram is a
rectangle, rhombus and square.
ANSWER:
Rectangle, rhombus, square; consecutive sides are
, all sides are .
6.Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are not perpendicular. So, it is not a
rhombus either.
ANSWER:
None; the diagonals are not congruent or
perpendicular.
ALGEBRA Quadrilateral ABCD is a rhombus.
Find each value or measure.
7.If AB = 14, find BC.
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So,
Therefore, BC = 14.
ANSWER:
14
8.If , find .
SOLUTION:
A rhombus is a parallelogram with all four sides
congruent. So, Then, isan
isosceles triangle. Therefore,
If a parallelogram is a rhombus, then each diagonal
bisects a pair of opposite angles. So,
Therefore,
ANSWER:
27
9.If AP = 3x – 1 and PC = x + 9, find AC.
SOLUTION:
Thediagonalsofarhombusbisecteachother.
3x – 1 = x + 9
2x = 10
x = 5
Therefore, AC = 2(5 + 9) = 28.
ANSWER:
28
10.If DB = 2x – 4 and PB = 2x – 9, find PD.
SOLUTION:
The diagonals of a rhombus bisect each other. So,
Therefore, PD = PB = 2(7) – 9 = 5.
ANSWER:
5
11.If , find
.
SOLUTION:
In a rhombus, consecutive interior angles are
supplementary.
Each pair of opposite angles of a rhombus is
congruent. So,
ANSWER:
95
12.If , find x.
SOLUTION:
The diagonals of a rhombus are perpendicular to
each other.
ANSWER:
35
CCSSARGUMENTSWriteatwo-column
proof.
13.Given:
Prove: WXYZ is a rhombus.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are
given . You
need to prove that WXYZ is a rhombus. Use the
properties that you have learned about parallelograms
and rhombi to walk through the proof.
Given:
Prove: WXYZ is a rhombus.
Proof:
Statements(Reasons)
1. (Given)
2. WXYZ is a . (Both pairs of opp. sides are .)
3. WXYZ is a rhombus. (If one pair of consecutive
sides of a are , the isarhombus.)
ANSWER:
Given:
Prove: WXYZ is a rhombus.
Proof:
Statements(Reasons)
1. (Given)
2. WXYZ is a . (Both pairs of opp. sides are .)
3. WXYZ is a rhombus. (If one pair of consecutive
sides of a are , the isarhombus.)
14.Given: QRST is a parallelogram.
Prove: QRST is a square.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove.Here,youaregivenQRST is a parallelogram;
. You need to prove that
QRST is a square. Use the properties that you have
learned about parallelograms and squares to walk
through the proof.
Given: QRST is a parallelogram;
Prove: QRST is a square.
Proof:
Statements(Reasons)
1. QRST is a parallelogram;
. (Given)
2. QRST is a rectangle. (If the diagonals of a
, the isarectangle.)
3. isarightangle.(Defofrt. )
4. (Def.ofperpendicular)
5. QRST is a rhombus. (If the diagonals of a
isarhombus.)
6. QRST is a square. (Thm.6.2, if a quadrilateral is a
rectangle and a rhombus, then it is a square.)
ANSWER:
Given: QRST is a parallelogram;
Prove: QRST is a square.
Proof:
Statements(Reasons)
1. QRST is a parallelogram; . (Given)
2. QRST is a rectangle. (If the diagonals of a
, the isarectangle.)
3. isarightangle.(Defofrt. )
4. (Def.ofperpendicular)
5. QRST is a rhombus. (If the diagonals of a
, isarhombus.)
6. QRST is a square. (Thm.6.2, if a quadrilateral is a
rectangle and a rhombus, then it is a square.)
15.Given: JKQP is a square. .
Prove: JKLM is a parallelogram.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove.Here,youaregiven
JKQP is a square and . You
need to prove that JKLM is a parallelogram. Use the
properties that you have learned about parallelograms
to walk through the proof.
Given: JKQP is a square.
.
Prove: JKLM is a parallelogram.
Proof:
Statements (Reasons)
1. JKQP is a square. .
(Given)
2. JKQP is a parallelogram. (All squares are
parallelograms.)
3. (Def.of )
4. (Opp.Sidesof .)
5. JP = KQ (Def of segs.)
6. JM = MP, KL = LQ (Def. of bisects)
7. JP = JM + MP, KQ = KL + LQ (Seg. Add Post.)
8. JP = 2JM, KQ = 2KL (Subst.)
9. 2JM = 2KL (Subst.)
10. JM = KL (Division Prop.)
11. (Def.of segs.)
12. JKLM is a parallelogram. (If one pair of opp.
sides is , then the quad. is a .)
ANSWER:
Proof:
Statements (Reasons)
1. JKQP is a square. .
(Given)
2. JKQP is a parallelogram. (All squares are
parallelograms.)
3. (Def.of )
4. (Opp.sidesofof .)
5. JP = KQ (Def of segs.)
6. JM = MP, KL = LQ (Def. of bisects)
7. JP = JM + MP, KQ = KL + LQ (Seg. Add Post.)
8. JP = 2JM, KQ = 2KL (Subst.)
9. 2JM = 2KL (Subst.)
10. JM = KL (Division Prop.)
11. (Def.of segs.)
12. JKLM is a parallelogram. (If one pair of opp.
sides is , then the quad. is a .)
16.Given: ACDH and BCDF are parallelograms;
.
Prove: ABFH is a rhombus.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ACDH and BCDF are
parallelograms; .You need to prove that
ABHF is a rhombus. Use the properties that you
have learned about parallelograms and rhombi to
walk through the proof.
Given: ACDH and BCDF are parallelograms;
.
Prove: ABHF is a rhombus.
Proof:
Statements (Reasons)
1. ACDH and BCDF are parallelograms; .
(Given)
2. (Def.of )
3. (Trans.Prop)
4. (Def.of )
5. AC = HD (Def of segs.)
6. AC = AB + BC, HD = HF + FD (Seg. Add. Post.)
7. AC – HD = AB + BC – HF – FD (Subt. Prop.)
8. AB = HF (Subst.)
9. (Def.of segs.)
10. (Subst.)
11. ABFH is a rhombus. (Def. of rhombus)
ANSWER:
Proof:
Statements (Reasons)
1. ACDH and BCDF are parallelograms; .
(Given)
2. (Def.of )
3. (Trans.Prop)
4. (Def.of )
5. AC = HD (Def of segs.)
6. AC = AB + BC, HD = HF + FD (Seg. Add. Post.)
7. AC – HD = AB + BC – HF – FD (Subt. Prop.)
8. AB = HF (Subst.)
9. (Def.of segs.)
10. (Subst.)
11. ABFH is a rhombus. (Def. of rhombus)
17.ROADWAYS Main Street and High Street intersect
as shown in the diagram. Each of the crosswalks is
the same length. Classify the quadrilateral formed by
the crosswalks. Explain your reasoning.
SOLUTION:
The two streets intersect at a 29 degree angle.
Analyze this first to determine the measure of an
angle of the quadrilateral. Then analyze the
information given about the lengths of the crosswalks
to categorize the quadrilateral formed.
The measure of the angle formed between the two
streets is 29, and vertical angles are congruent, so
the measure of one angle of the quadrilateral is 29.
So the quadrilateral is not a rectangle or square.
Since the crosswalks are the same length, the sides
of the quadrilateral are congruent. Therefore, they
form a rhombus.
ANSWER:
Rhombus; Sample answer: The measure of angle
formed between the two streets is 29, and vertical
angles are congruent, so the measure of one angle of
the quadrilateral is 29. Since the crosswalks are the
same length, the sides of the quadrilateral are
congruent. Therefore, they form a rhombus.
18.CCSS MODELING A landscaper has staked out
the area for a square garden as shown. She has
confirmed that each side of the quadrilateral formed
by the stakes is congruent and that the diagonals are
perpendicular. Is this information enough for the
landscaper to be sure that the garden is a square?
Explain your reasoning.
SOLUTION:
Compare the information given to the criteria to
classify a quadrilateral as a square.
Since the four sides of the quadrilateral are
congruent and the diagonals are perpendicular, the
figure is either a square or a rhombus. To be sure
that the garden is a square, she also needs to confirm
that the diagonals are congruent.
ANSWER:
No; sample answer: Since the four sides of the
quadrilateral are congruent and the diagonals are
perpendicular, the figure is either a square or a
rhombus. To be sure that the garden is a square, she
also needs to confirm that the diagonals are
congruent.
COORDINATE GEOMETRY Given each set
of vertices, determine whether JKLM is a
rhombus, a rectangle, or a square. List all that
apply. Explain.
19.J(–4, –1), K(1, –1), L(4, 3), M(–1, 3)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are perpendicular. So, it is a rhombus.
ANSWER:
Rhombus; the diagonals are .
20.J(–3, –2), K(2, –2), L(5, 2), M(0, 2)
SOLUTION:
First, graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are perpendicular. So, it is a rhombus.
ANSWER:
Rhombus; the diagonals are .
21.J(–2, –1), K(–4, 3), L(1, 5), M(3, 1)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are not congruent. So, the
parallelogram is not a rectangle. Check whether the
two diagonals are perpendicular.
The diagonals are not perpendicular. So, it is not a
rhombus either.
ANSWER:
None; the diagonals are not .
22.J(–1, 1), K(4, 1), L(4, 6), M(–1, 6)
SOLUTION:
First graph the quadrilateral.
If the diagonals of the parallelogram are congruent,
then it is a rectangle. Use the Distance Formula to
find the lengths of the diagonals.
The diagonals are congruent. So, the parallelogram is
a rectangle. Check whether the two diagonals are
perpendicular.
The diagonals are perpendicular. So, it is a rhombus.
Since the diagonals are both congruent and
perpendicular to each other the parallelogram is a
rectangle, rhombus and square.
ANSWER:
Square, rectangle, rhombus; all sides are .
ABCD is a rhombus. If PB = 12, AB = 15, and
, find each measure.
23.AP
SOLUTION:
The diagonals of a rhombus are perpendicular to
each other. So, by the Pythagorean Theorem, AP2 =
AB2 – PB2.
ANSWER:
9
24.CP
SOLUTION:
All the four sides of a rhombus are congruent and the
diagonalsareperpendiculartoeachother.
So, by the Pythagorean Theorem, CP2 = BC2 – PB2.
BC = AB. Substitute AB for BC.
ANSWER:
9
25.
SOLUTION:
All the four sides of a rhombus are congruent. So,
isanisoscelestriangle.Then,
ANSWER:
24
26.
SOLUTION:
The diagonals are perpendicular to each other. So, in
the right triangle PAB,
All the four sides of a rhombus are congruent. So,
isanisoscelestriangle.Then,
ANSWER:
66
WXYZ is a square. If WT = 3, find each
measure.
27.ZX
SOLUTION:
The diagonals of a square are congruent and bisect
each other.
So, ZX = WY = 2(WT) = 6.
ANSWER:
6
28.XY
SOLUTION:
The diagonals of a square are congruent and bisect
each other at right angles.
So, YT = XT = WT = 3.
By the Pythagorean Theorem, XY2 = YT2 + XT2.
ANSWER:
29.
SOLUTION:
The diagonals of a square are perpendicular to each
other.
So,
ANSWER:
90
30.
SOLUTION:
In a square, each diagonal bisects a pair of opposite
angles. So,
ANSWER:
45
Classify each quadrilateral.
31.Refer to the photo on p. 432.
SOLUTION:
The diagonals are congruent and bisect each other at
right angle. Therefore, the quadrilateral is a square.
ANSWER:
square
32.Refer to the photo on p. 432.
SOLUTION:
The two pairs of opposite angles are congruent and
the adjacent sides are congruent. Therefore, the
quadrilateral is a rhombus.
ANSWER:
rhombus
33.Refer to the photo on p. 432.
SOLUTION:
The two pairs of opposite sides are congruent and
one of the angles is a right angle. Therefore, the
quadrilateral is a rectangle.
ANSWER:
rectangle
PROOF Write a paragraph proof.
34.Theorem 6.16
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rhombus. You
need to prove that each diagonal bisects a pair of
opposite angles.. Use the properties that you have
learned about rhombi to walk through the proof.
Given: ABCD is a rhombus.
Prove: Each diagonal bisects a pair of opposite
angles.
Proof: We are given that ABCD is a rhombus. By
definition of rhombus, ABCD is a parallelogram.
Opposite angles of a parallelogram are congruent, so
.
becauseallsidesofarhombus
are congruent. bySAS.
byCPCTC.
bySAS. by
CPCTC. By definition of angle bisector, each
diagonal bisects a pair of opposite angles.
ANSWER:
Given: ABCD is a rhombus.
Prove: Each diagonal bisects a pair of opposite
angles.
Proof: We are given that ABCD is a rhombus. By
definition of rhombus, ABCD is a parallelogram.
Opposite angles of a parallelogram are congruent, so
.
becauseallsidesofarhombus
are congruent. bySAS.
byCPCTC.
bySAS. by
CPCTC. By definition of angle bisector, each
diagonal bisects a pair of opposite angles.
35.Theorem 6.17
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a parallelogram;
. You need to prove that ABCD is a
rhombus. Use the properties that you have learned
about rhombi to walk through the proof.
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: We are given that ABCD is a parallelogram.
The diagonals of a parallelogram bisect each other,
so becausecongruenceof
segments is reflexive. We are also given that
. Thus, areright
angles by the definition of perpendicular lines. Then
becauseallrightanglesare
congruent. Therefore, bySAS.
byCPCTC.Oppositesidesof
parallelograms are congruent, so
. Then since congruence of
segments is transitive, . All
four sides of ABCD are congruent, so ABCD is a
rhombus by definition.
ANSWER:
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: We are given that ABCD is a parallelogram.
The diagonals of a parallelogram bisect each other,
so becausecongruenceof
segments is reflexive. We are also given that
. Thus, areright
angles by the definition of perpendicular lines. Then
becauseallrightanglesare
congruent. Therefore, bySAS.
byCPCTC.Oppositesidesof
parallelograms are congruent, so
. Then since congruence of
segments is transitive, . All
four sides of ABCD are congruent, so ABCD is a
rhombus by definition.
36.Theorem 6.18
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a diagonal of a
parallelogram bisects an angle of a parallelogram.
You need to prove that the parallelogram is a
rhombus. Use the properties that you have learned
about parallelograms and rhombi to walk through the
proof.
Given: ABCD is a parallelogram; diagonal
bisects .
Prove: ABCD is a rhombus.
Proof: It is given that ABCD is a parallelogram. Since
opposite sides of a parallelogram are parallel,
. By definition, arealternate
interior angles of parallel sides . Since
alternate interior angles are congruent, .
Congruence of angles is symmetric, therefore
. it is given that bisects
, so by
definition. By the Transitive Property,
.
The sides opposite congruent angles in a triangle are
congruent, therefore, . So,
since a pair of consecutive sides of the parallelogram
is congruent, ABCD is a rhombus.
ANSWER:
If a diagonal of a parallelogram bisects an angle of a
parallelogram, then the parallelogram is a rhombus.
Given: ABCD is a parallelogram; diagonal
bisects .
Prove: ABCD is a rhombus.
Proof: It is given that ABCD is a parallelogram. Since
opposite sides of a parallelogram are parallel,
. By definition, arealternate
interior angles of parallel sides . Since
alternate interior angles are congruent, .
Congruence of angles is symmetric, therefore
. it is given that bisects
, so by
definition. By the Transitive Property,
.
The sides opposite congruent angles in a triangle are
congruent, therefore, . So,
since a pair of consecutive sides of the parallelogram
is congruent, ABCD is a rhombus.
37.Theorem 6.19
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a parallelogram;
. You need to prove that ABCD is a
rhombus. Use the properties that you have learned
about rhombi to walk through the proof.
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: Opposite sides of a parallelogram are
congruent, so . We are
given that . So, by the Transitive Property,
. So, . Thus, ABCD
is a rhombus by definition.
ANSWER:
Given: ABCD is a parallelogram; .
Prove: ABCD is a rhombus.
Proof: Opposite sides of a parallelogram are
congruent, so . We are
given that . So, by the Transitive Property,
. So, . Thus, ABCD
is a rhombus by definition.
38.Theorem 6.20
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rectangle and
a rhombus. You need to prove that ABCD is a
square. Use the properties that you have learned
about squares to walk through the proof.
Given: ABCD is a rectangle and a rhombus.
Prove: ABCD is a square.
Proof: We know that ABCD is a rectangle and a
rhombus. ABCD is a parallelogram, since all
rectangles and rhombi are parallelograms. By the
definition of a rectangle, are
right angles. By the definition of a rhombus, all of the
sides are congruent. Therefore, ABCD is a square
since ABCD is a parallelogram with all four sides
congruent and all the angles are right.
ANSWER:
Given: ABCD is a rectangle and a rhombus.
Prove: ABCD is a square.
Proof: We know that ABCD is a rectangle and a
rhombus. ABCD is a parallelogram, since all
rectangles and rhombi are parallelograms. By the
definition of a rectangle, are
right angles. By the definition of a rhombus, all of the
sides are congruent. Therefore, ABCD is a square
since ABCD is a parallelogram with all four sides
congruent and all the angles are right.
CONSTRUCTION Use diagonals to construct
each figure. Justify each construction.
39.rhombus
SOLUTION:
Sample answer: Construct the perpendicular bisector
of a line segment. Place the compass at the midpoint
of the segment. Use the same compass setting to
locate a point on the perpendicular bisector above
and below the segment. Connect the endpoints of the
segments with the two points on the perpendicular
bisectortoformaquadrilateral.
Since the diagonals bisect each other, the
quadrilateral is a parallelogram. If the diagonals of a
parallelogram are perpendicular to each other, then
the parallelogram is a rhombus. Thus, the constructed
quadrilateral is a rhombus.
ANSWER:
Sample answer: The diagonals bisect each other, so
the quadrilateral is a parallelogram. Since the
diagonals of the parallelogram are perpendicular to
each other, the parallelogram is a rhombus.
40.square
SOLUTION:
Sample answer: Construct the perpendicular bisector
of a segment. Adjust the compass setting to equal the
distance from the midpoint of the segment to one of
its endpoints. Place the compass at the midpoint of
the segment. Draw arcs that intersect the
perpendicular bisector above and below the segment.
Connect the two points of intersection with the
endpoints of the segment to form a quadrilateral.
Since the diagonals bisect each other, the
quadrilateral is a parallelogram. If the diagonals of a
parallelogram are congruent and perpendicular, then
the parallelogram is a square. Thus, the constructed
quadrilateral is a square.
ANSWER:
Sample answer: The diagonals bisect each other, so
the quadrilateral is a parallelogram. Since the
diagonals of the parallelogram are congruent and
perpendicular, the parallelogram is a square.
PROOF Write a coordinate proof of each
statement.
41.The diagonals of a square are perpendicular.
SOLUTION:
Begin by positioning square ABCD on a coordinate
plane. Place vertex A at the origin. Let the length of
the bases be a units. Then the rest of the vertices are
B(a, 0), C(a, a), and D(0, a). You need to walk
through the proof step by step. Look over what you
are given and what you need to prove. Here, you are
given ABCD is a square and you need to prove that
. Use the properties that you have learned
about squares to walk through the proof.
Given: ABCD is a square.
Prove:
Proof:
slope of
slope of
The slope of is the negative reciprocal of the
slope of , so they are perpendicular.
ANSWER:
Given: ABCD is a square.
Prove:
Proof:
slope of
slope of
The slope of is the negative reciprocal of the
slope of , so they are perpendicular.
42.The segments joining the midpoints of the sides of a
rectangle form a rhombus.
SOLUTION:
Begin by positioning rectangle ABCD on a coordinate
plane. Place vertex A at the origin. Let the length of
the bases be a units and the height be b units. Then
the rest of the vertices are B(a, 0), C(a, b), and D(0,
b). You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rectangle and
Q, R, S, and T are midpoints of their respective
sides.and you need to prove that QRST is a rhombus.
Use the properties that you have learned about
rhombi to walk through the proof.
Given: ABCD is a rectangle. Q, R, S, and T are
midpoints of their respective sides.
Prove: QRST is a rhombus.
Proof:
Midpoint Q is .
Midpoint R is .
Midpoint S is .
Midpoint T is .
QR = RS = ST = QT
QRST is a rhombus.
ANSWER:
Given: ABCD is a rectangle. Q, R, S, and T are
midpoints of their respective sides.
Prove: QRST is a rhombus.
Proof:
Midpoint Q is .
Midpoint R is .
Midpoint S is .
Midpoint T is .
QR = RS = ST = QT
QRST is a rhombus.
43.DESIGN The tile pattern below consists of regular
octagons and quadrilaterals. Classify the
quadrilaterals in the pattern and explain your
reasoning.
SOLUTION:
In order to classify the quadrilaterals we need
information about the interior angles and the sides.
It's given that each quadrilateral is formed by 4
regular octagons. We can use what we know about
the exterior angles of a regular octagon as well as
the sides of a regular octagon to determine which
type of quadrilateral is in the pattern.
Squares; sample answer: Since the octagons are
regular each side is congruent, and the quadrilaterals
share common sides with the octagon, so the
quadrilaterals are either rhombi or squares. The
vertices of the quadrilaterals are formed by the
exterior angles of the sides of the octagons adjacent
to the vertices. The sum of the measures of the
exterior angles of a polygon is always 360 and since
a regular octagon has 8 congruent exterior angles,
each one measures 45. As shown in the diagram,
each angle of the quadrilaterals in the pattern
measures 45 + 45 or 90. Therefore, the quadrilateral
is a square.
ANSWER:
Squares; sample answer: Since the octagons are
regular each side is congruent, and the quadrilaterals
share common sides with the octagon, so the
quadrilaterals are either rhombi or squares. The
vertices of the quadrilaterals are formed by the
exterior angles of the sides of the octagons adjacent
to the vertices. The sum of the measures of the
exterior angles of a polygon is always 360 and since
a regular octagon has 8 congruent exterior angles,
each one measures 45. As shown in the diagram,
each angle of the quadrilaterals in the pattern
measures 45 + 45 or 90. Therefore, the quadrilateral
is a square.
44.REPAIR The window pane shown needs to be
replaced. What are the dimensions of the
replacement pane?
SOLUTION:
The window pane is in the shape of a square. The
diagonal of a square is the hypotenuse of a right
triangle with two consecutive sides of the square as
its legs. Let x be the length of each side of the
square. So, by the Pythagorean Theorem,
Therefore, the length of each side of the square is
about 15 inches.
ANSWER:
square; 15 in.
45.MULTIPLE REPRESENTATIONS In this
problem, you will explore the properties of kites,
which are quadrilaterals with exactly two distinct
pairs of adjacent congruent sides.
a. GEOMETRIC Draw three kites with varying
side lengths. Label one kite ABCD, one PQRS, and
one WXYZ. Then draw the diagonals of each kite,
labeling the point of intersection N for each kite.
b. TABULAR Measure the distance from N to
each vertex. Record your results in a table like the
one shown.
c. VERBAL Make a conjecture about the diagonals
of a kite.
SOLUTION:
a. Sample answer: Draw three different kites, each
with the intersection of the diagonals labeled N.
b.Use a ruler to measure each length in the table.
c. Sample answer: From the measurements recorded
in the table, N is the midpoint of each of the short
diagonals. The shorter diagonal of a kite is bisected
by the longer diagonal.
ANSWER:
a. Sample answer:
b.
c. Sample answer: The shorter diagonal of a kite is
bisected by the longer diagonal.
46.ERROR ANALYSIS In quadrilateral PQRS,
. Lola thinks that the quadrilateral is a
square, and Xavier thinks that it is a rhombus. Is
either of them correct? Explain your reasoning.
SOLUTION:
The only information known is that the diagonals are
congruent. Review the quadrilaterals that have
congruent diagonals. Is this enough information to
classify the quadrilateral?
Since they do not know that the sides of the
quadrilateral are congruent, only that the diagonals
are congruent, they can only conclude that the
quadrilateral is a rectangle. So, neither of them are
correct.
ANSWER:
Neither; sample answer: Since they do not know that
the sides of the quadrilateral are congruent, only that
the diagonals are congruent, they can only conclude
that the quadrilateral is a rectangle.
47.CCSS ARGUMENTS Determine whether the
statement is true or false. Then write the converse,
inverse, and contrapositive of the statement and
determine the truth value of each. Explain your
reasoning.
If a quadrilateral is a square, then it is a
rectangle.
SOLUTION:
True; sample answer: A rectangle is a quadrilateral
with four right angles and a square is both a
rectangle and a rhombus, so a square is always a
rectangle.
Converse: If a quadrilateral is a rectangle then it is a
square. False; sample answer: A rectangle is a
quadrilateral with four right angles. It is not
necessarily a rhombus, so it is not necessarily a
square.
Inverse: If a quadrilateral is not a square, then it is
not a rectangle. False; sample answer: A
quadrilateral that has four right angles and two pairs
of congruent sides is not a square, but it is a
rectangle.
Contrapositive: If a quadrilateral is not a rectangle,
then it is not a square. True; sample answer: If a
quadrilateral is not a rectangle, it is also not a square
by definition.
ANSWER:
True; sample answer: A rectangle is a quadrilateral
with four right angles and a square is both a
rectangle and a rhombus, so a square is always a
rectangle.
Converse: If a quadrilateral is a rectangle then it is a
square. False; sample answer: A rectangle is a
quadrilateral with four right angles. It is not
necessarily a rhombus, so it is not necessarily a
square.
Inverse: If a quadrilateral is not a square, then it is
not a rectangle. False; sample answer: A
quadrilateral that has four right angles and two pairs
of congruent sides is not a square, but it is a
rectangle.
Contra positive: If a quadrilateral is not a rectangle,
then it is not a square. True; sample answer: If a
quadrilateral is not a rectangle, it is also not a square
by definition.
48.CHALLENGE The area of square ABCD is 36
square units and the area of is 20 square
units.If , find the length of
.
SOLUTION:
Since the area of the square is 36 square units, the
length of each side of the square is 6 units. All the
four angles of a square are right angles. So, by the
Pythagorean Theorem,
The area of is20squareunits.So,
Also, we have
So, by HL postulate,
AE = CF by CPCTC
Therefore, CF = 2.
ANSWER:
2
49.OPEN ENDED Find the vertices of a square with
diagonals that are contained in the lines y = x and y =
−x + 6. Justify your reasoning.
SOLUTION:
Sampleanswer:
First graph the lines y = x and y = -x + 6. There are 6
units from the origin to the y-intercept of y = -x + 6
and 6 units from the origin to the x-intercept of y = -x
+ 6. So, three of the vertices of a square will be at (0,
0), ((0, 6), and (6, 0). The point 6 units above (6, 0) is
(6, 6).
(0, 0), (6, 0), (0, 6), (6, 6); the diagonals are
perpendicular, and any four points on the lines
equidistant from the intersection of the lines will be
the vertices of a square.
ANSWER:
Sample answer: (0, 0), (6, 0), (0, 6), (6, 6); the
diagonals are perpendicular, and any four points on
the lines equidistant from the intersection of the lines
will be the vertices of a square.
50.WRITING IN MATH Compare all of the propertie
rectangles, rhombi, and squares.
SOLUTION:
Sample answer:
Parallelogram: Opposite sides of a parallelogram are
parallelogram are congruent. The diagonals of a paral
separates a parallelogram into two congruent triangle
Rectangle: A rectangle has all the properties of a par
The diagonals of a rectangle are congruent.
Rhombus: A rhombus has all of the properties of a pa
congruent. The diagonals of a rhombus are perpendic
Square: A square has all of the properties of a paralle
rectangle. A square has all of the properties of a rho
ANSWER:
Sample answer:
Parallelogram: Opposite sides of a parallelogram are
parallelogram are
congruent. The diagonals of a parallelogram bisect ea
parallelogram into two congruent triangles.
Rectangle: A rectangle has all the properties of a par
The diagonals of a rectangle are congruent.
Rhombus: A rhombus has all of the properties of a pa
congruent. The diagonals of a rhombus are perpendic
Square: A square has all of the properties of a paralle
rectangle. A square has all of the properties of a rho
51.JKLM is a rhombus. If CK = 8 and JK = 10, find JC.
A 4C 8
B 6D 10
SOLUTION:
The diagonals of a rhombus are perpendicular to
each other. So, by the Pythagorean Theorem,
JC2 = JK2 – CK2.
Therefore, the correct choice is B.
ANSWER:
B
52.EXTENDED RESPONSE The sides of square
ABCD are extended by sides of equal length to form
square WXYZ.
a. If CY = 3 cm and the area of ABCD is 81 cm2,
find the area of WXYZ.
b. If the areas of ABCD and WXYZ are
respectively, find DZ.
c. If AB = 2CY and the area of ABCD = g square
meters, find the area of WXYZ in square meters.
SOLUTION:
a. Since the area of the square is 81 square cm, the
length of each side of the square is 9 cm. Since the
sides of the square ABCD are extended to form
square WXYZ, DZ = AW = BX = CY = 3 cm. Area of
each of the triangle is
The area of the square WXYZ is the sum of the 4
congruent triangles and the area of the square.
b. In the right triangle WZD, WZ = 13 and WD = AW
+ AD = DZ + 7 = x + 7 where x is the length of the
segment BythePythagoreanTheorem,
Solve the equation for x.
Since x is a length, it cannot be negative. Therefore,
the length DZ is about 5 cm.
c. If AB = 2CY, the lengths of the two legs of the
are
are
,
are
are
are
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6-5 Rhombi and Squares
