In the figure, m1 = 94. Find the measure of
each angle. Tell which postulate(s) or theorem
(s) you used.
1. 3
SOLUTION:
In the figure, angles 1 and 3 are corresponding
angles. Use the Corresponding Angles Postulate.
If two parallel lines are cut by a transversal, then
each pair of corresponding angles are congruent.
ANSWER:
94; Corresponding Angle Postulate
2. 5
SOLUTION:
In the figure, angles 1 and 5 are alternate exterior
angles.
Use the Alternate Exterior Angles Theorem.
If two parallel lines are cut by a transversal, then
each pair of alternate interior angles is congruent.
ANSWER:
94; Alt. Ext. s Thm.
3. 4
SOLUTION:
In the figure, angles 1 and 3 are corresponding
angles. Use the Corresponding Angles Postulate:
If two parallel lines are cut by a transversal, then
each pair of corresponding angles is congruent.
ANSWER:
86; Corresponding Angle Postulate and Supplement
Angle Thm.
In the figure, m4 = 101. Find the measure of
each angle. Tell which postulate(s) or theorem
(s) you used.
4. 6
SOLUTION:
In the figure, angles 4 and 6 are alternate interior
angles.
Use the Alternate interior Angles Theorem.
If two parallel lines are cut by a transversal, then
each pair of alternate interior angles is congruent.
ANSWER:
101; Alt. Int. s Thm.
5. 7
SOLUTION:
In the figure, angles 4 and 5 are consecutive interior
angles.
In the figure, angles 5 and 7 are vertical angles.
ANSWER:
79; Vertical Angle Thm. Cons. Int. s Thm.
6. 5
SOLUTION:
In the figure, angles 4 and 5 are consecutive interior
angles.
ANSWER:
79; Cons. Int. s Thm.
7.ROADS In the diagram, the guard rail is parallel to
the surface of the roadway and the vertical supports
are parallel to each other. Find the measures of
angles 2, 3, and 4.
SOLUTION:
Use the Alternate Interior Angles Theorem,
Definition of Supplementary Angles and
Corresponding Angles Postulate to find m 4.
So,m 4 = 87.
ANSWER:
m2 = 93, m3 = 87, m4 = 87
Find the value of the variable(s) in each figure.
Explain your reasoning.
8.
SOLUTION:
Use the definition of supplementary angles to find
. Then use the Alternate Interior Angles
Theorem to find .
ANSWER:
x = 125 by the Supplement Thm.;
y = 125 by the Alt. Int. s Thm.
9.
SOLUTION:
Use the Alternate Exterior Angles Theorem to find
x.
ANSWER:
x = 114 by the Alt. Ext. s Thm.
10.
SOLUTION:
Use the Alternate Interior Angles Theorem to find x.
ANSWER:
x = 70 by the Alt. Int. s Thm.
In the figure, m11 = 62 and m14 = 38. Find
the measure of each angle. Tell which postulate
(s) or theorem(s) you used.
11. 4
SOLUTION:
In the figure, angles 4 and 11 are corresponding
angles.
ANSWER:
62; Corr. s Post.
12. 3
SOLUTION:
In the figure, angles 4 and 11 are corresponding
angles and angles 3 and 4 are vertical angles.
ANSWER:
62; Corresponding s Post. and Vertical Thm.or
Alt. Ext. Thm.
13. 12
SOLUTION:
In the figure, angles 12 and 11 are supplementary
angles.
ANSWER:
118; Def. Supp. s
14. 8
SOLUTION:
In the figure, angles 8 and 11 are vertical angles.
ANSWER:
62; Vertical Angle Thm.
15. 6
SOLUTION:
In the figure, angles 14 and 6 are corresponding
angles.
ANSWER:
38; Corr. s Post.
16. 2
SOLUTION:
The angles 1 and 14 are alternate exterior
angles and so are congruent. and angles 3 and
11 are alternate exterior angles and so are
congruent. By Supplementary Theorem, m1 + m
2 + m3 = 180.
ANSWER:
80; Alt. Ext. s Post. and Supp. Thm.
17. 10
SOLUTION:
In the figure, angles 14 and 10 are supplementary
angles.
ANSWER:
142; Supplement Angles Thm.
18. 5
SOLUTION:
Use definition of supplementary angles,
Corresponding Angles Postulate and the Alternate
Interior Angles Theorem .
ANSWER:
80; Vertical Angles Thm.
19. 1
SOLUTION:
In the figure, angles 1 and 14 are alternate exterior
angles.
ANSWER:
38; Alt. Ext. s Thm.
CCSS MODELING A solar dish collects energy
by directing radiation from the Sun to a receiver
located at the focal point of the dish. Assume
that the radiation rays are parallel. Determine
the relationship between each pair of angles and
explain your reasoning.
Refer to Page 183.
20. 1 and 2
SOLUTION:
If the radiation rays form parallel lines, then ∠1 and
∠2 are consecutive interior angles. So, according to
the Consecutive Interior Angles Theorem, ∠1 and
∠2 are supplementary.
ANSWER:
supplementary; Consecutive Interior Angles
21. 1 and 3
SOLUTION:
If the radiation rays form parallel lines, then ∠1 and
∠3 are corresponding angles. So, according to the
Corresponding Angles Postulate, ∠1 and ∠3 are
congruent.
ANSWER:
congruent; Corresponding Angles
22. 4 and 5
SOLUTION:
If the radiation rays form parallel lines, then ∠4 and
∠5 are alternate exterior angles. So, according to the
Alternate Exterior Angles Theorem, ∠4 and ∠5 are
congruent.
ANSWER:
congruent; Alternate Exterior Angles
23. 3 and 4
SOLUTION:
If the radiation rays form parallel lines, then ∠3 and
∠5 are a linear pair of angles.. So, according to the
definition of linear pairs, ∠3 and ∠5 are
supplementary.∠4 and ∠5 are alternate exterior
angles. So, by the Alternate Exterior Angles
Theorem, ∠4 is congruent to ∠5. Then,
Therefore, ∠3 is supplementary to ∠4 by the
definition of supplementary angles.
ANSWER:
supplementary; since 3 and 5 are a linear pair,
they are supplementary. 4 and 5 are congruent
because they are alternate exterior angles, so 3 is
supplementary to 4.
Find the value of the variable(s) in each figure.
Explain your reasoning.
24.
SOLUTION:
Use Corresponding Angles Postulate and definition
of supplementary angles to find x.
ANSWER:
y = 114 by the Corresponding Angles Postulate; x =
54 by the Supplement Theorem
25.
SOLUTION:
Use the Corresponding Angles Postulate and
Supplement Theorem to find x and y.
ANSWER:
x = 40 by the Corresponding Angles Postulate; y =
50 by the Supplement Theorem
26.
SOLUTION:
Use the Vertical Angle Theorem and Consecutive
Interior Angles Theorem to find x.
ANSWER:
x = 63 by the Vertical Angle Theorem and the
Consecutive Interior Angles Theorem
27.
SOLUTION:
Use the Consecutive Interior Angles Theorem to find
x and y.
ANSWER:
x = 42 by the Consecutive Interior Angles Theorem;
y = 14 by the Consecutive Interior Angles Theorem
28.
SOLUTION:
Use the Alternate Interior Angles Theorem and
Consecutive Interior Angles Theorem to find x and y.
ANSWER:
x = 54 by the Alternate Interior Angles Theorem; y =
12 by the Consecutive Interior Angles Theorem
29.
SOLUTION:
Use the Consecutive Interior Angles Theorem
anddefinitionofsupplementaryanglestofind x and
y.
ANSWER:
x = 60 by the Consecutive Interior Angles Theorem;
y = 10 by the Supplement Theorem
30.PROOF Copy and complete the proof of Theorem
3.2.
Given: isatransversal.
Prove: 1 and 2 are supplementary;
3 and 4 are supplementary.
Proof:
SOLUTION:
ANSWER:
STORAGE When industrial shelving needs to
be accessible from either side, additional
support is provided on the side by transverse
members. Determine the relationship between
each pair of angles and explain your reasoning.
31. 1 and 8
SOLUTION:
1 and 8 are Alternate interior angles. Therefore
1 and 8 are congruent.
ANSWER:
Congruent; Alternate interior angles
32. 1 and 5
SOLUTION:
1 and 5 are Corresponding angles. Therefore,
they are congruent.
ANSWER:
Congruent; Corresponding angles
33. 3 and 6
SOLUTION:
3 and 6 are Vertical angles. Therefore Vertical
angles are congruent.
ANSWER:
Congruent; Vertical angles are congruent
34. 1 and 2
SOLUTION:
All vertical and horizontal lines are perpendicular at
their point of intersection.By definition of
perpendicular,theyformrightangles.∠1 and ∠2
are adjacent angles. By the Angle Addition Postulate,
m∠1+m∠2 = 90. Since the sum of the two angles is
90, ∠1 and ∠2 are complementary angles.
ANSWER:
Complementary; because the vertical and horizontal
lines are perpendicular, they form right angles.
35.CCSS ARGUMENTS Write a two-column proof of
the Alternate Exterior Angles Theorem.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. (Corr. s Post.)
3. (Vertical s Thm.)
4. (Trans.Prop.)
ANSWER:
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. 1 5, 2 6 (Corr. s Post.)
3. 5 8, 6 7 (Vertical s Thm.)
4. 1 8, 2 7 (Trans. Prop.)
36.BRIDGES Refer to the diagram of the double
decker Michigan Avenue Bridge in Chicago, Illinois.
The two levels of the bridge, and its diagonal braces,
are parallel.
a. How are the measures of the odd-numbered
angles related? Explain.
b. How are the measures of the even-numbered
angles related? Explain.
c. How are any pair of angles in which one is odd
and the other is even related?
d. What geometric term(s) can be used to relate the
two roadways contained by the bridge?
SOLUTION:
a. The top and bottom levels of the bridge are
parallel, so the lines formed by the edge of each level
are parallel, and by using the diagonal braces as
transversals and the Alternate Interior Angles
Theorem ∠1 ∠3, ∠5 ∠7, ∠9 ∠11, and
∠13 ∠15.
The diagonal braces are parallel, so by using the
vertical braces as transversals and the Alternate
Interior Angles Theorem ∠4 ∠6, ∠8 ∠10,
and ∠12 ∠14.
Since the vertical braces are perpendicular to the
levels of the bridge, ∠3 and ∠4, ∠5 and ∠6, ∠7 and
∠8, ∠9 and ∠10, ∠11 and ∠12, and ∠13 and ∠14
arepairsofcomplementaryangles.
By the Congruent Complements Theorem, ∠3
∠5, ∠7 ∠9, and ∠11 ∠13. So, ∠1
∠3 ∠5 ∠7 ∠9 ∠11 ∠13 ∠13
by the Transitive Property of Congruence. So, all of
the odd numbered angles are alternate interior angles
related by the diagonal transversals or are
complements of even numbered alternate interior
angles related by the vertical transversals. Therefore,
they are all congruent.
b. All the vertical braces are parallel since all vertical
lines are parallel. Using the diagonal braces as
transversals to the vertical braces and the Alternate
Interior Angles Theorem, ∠2 ∠4, ∠6 ∠8,
∠10 ∠12, and ∠14 ∠16.
Using the vertical braces as transversals between the
diagonal braces and the Alternate Interior Angles
Theorem, ∠4 ∠6, ∠8 ∠10, and ∠12
∠14. So, ∠2 ∠4 ∠6 ∠8 ∠10
∠12 ∠14 ∠16 by the Transitive Property of
Congruence.
All of the even numbered angles are alternate interior
angles related by either the diagonal transversals or
the vertical transversals. Therefore, they are all
congruent.
c. Complementary; since the vertical supports and
the horizontal supports are perpendicular, angle pairs
like angle 1 and angle 2 must be complementary.
Since all of the odd numbered angles are congruent
and all of the even numbered angles are congruent,
any pair of angles that has one odd and one even
number will be complementary.
d. Since the two levels (or surfaces) of the bridge
are parallel, the geometric term that best represents
the two roadways contained by the bridge is parallel
planes.
ANSWER:
a. Congruent; all of the odd numbered angles are
alternateinterioranglesrelatedbythediagonal
transversals or are complements of even numbered
alternate interior angles related by the vertical
transversals, so they are all congruent.
b. Congruent; all of the even numbered angles are
alternate interior angles related by either the diagonal
transversals or the vertical transversals, so they are
all congruent.
c. Complementary; since the vertical supports and
the horizontal supports are perpendicular, angle pairs
like 1 and 2 must be complementary. Since all
of the odd numbered angles are congruent and all of
the even numbered angles are congruent, any pair of
angles that has one odd and one even number will be
complementary.
d. parallel planes
37.PROOF In a plane, prove that if a line is
perpendicular to one of two parallel lines, then it is
perpendicular to the other.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. Angle 1 is a right angle. (Def. of )
3. (Def.ofrt. )
4. (Corr. Post.)
5. (Def.of )
6. (Subs.)
7. isarightangle.(Def.ofrt. )
8. (Def.of lines)
ANSWER:
Given:
Prove:
Proof:
Statements (Reasons)
1. m || n, t ⊥ m (Given)
2. 1 is a right angle. (Def. of ⊥)
3. m1 = 90 (Def. of rt. s)
4. 1 2 (Corr. s Post.)
5. m1 = m2 (Def. of s)
6. m2 = 90 (Subs.)
7. 2 is a right angle. (Def. of rt. s)
8. t ⊥ n (Def. of ⊥ lines)
CCSSTOOLSFindx. (Hint: Draw an auxiliary
line.)
38.
SOLUTION:
Draw an auxiliary line to construct a triangle.
Then label the angles a°,b°,andc°.Byfindingthe
measures for angles a and b, we can use the
Triangle Angle Sum theorem to find angle c. Angles
c and xareverticalangles.
Use the definition of supplementary angles to find a.
Find angle b.
Find angle c.
Find angle x.
So, x = 22.
ANSWER:
22
39.
SOLUTION:
Draw an auxiliary line to construct a triangle.By
creating a triangle, we can sue the Triangle
Angle Sum Theorem and definition of
supplementary angles to find x. Label the
angles.
First find angle a.
Find angle b.
Find angle c.
Find angle d.
Find angle x
.
So x = 130º.
ANSWER:
130
40.PROBABILITY Suppose you were to pick any two
angles in the figure below.
a. How many possible angle pairings are there?
Explain.
b. Describe the possible relationships between the
measures of the angles in each pair. Explain.
c. Describe the likelihood of randomly selecting a
pair of congruent angles. Explain your reasoning.
SOLUTION:
a. Sample answer: There are 28 possible angle
pairings. The first angle can be paired with seven
others, then the second angle can be paired with six
others since it has already been paired with the first
angle. The number of pairings is the sum of the
number of angles each subsequent angle can be
paired with, 7 + 6 + 5 + 4 + 3 + 2 + 1 or 28 pairings.
b. Sample answer: Because the two lines being
transversed are parallel, there are only two possible
relationships between the pairs of angles. Each pair
of angles chosen will be either congruent or
supplementary.
Congruent pairs: ∠1 and ∠3, ∠1 and ∠5, ∠1 and
∠7, ∠3 and ∠5, ∠3 and ∠7, ∠5 and ∠7, ∠2 and
∠4, ∠2 and ∠6, ∠2 and ∠8, ∠4 and ∠6, ∠4 and
∠8, ∠6 and ∠8
Supplementary pairs: ∠1 and ∠2, ∠1 and ∠4, ∠1
and ∠6, ∠1 and ∠8, ∠2 and ∠3, ∠2 and ∠5, ∠2
and ∠7, ∠3 and ∠4, ∠3 and 6, ∠3 and ∠8, ∠4 and
∠5, ∠4 and ∠7, ∠5 and ∠6, ∠5 and ∠8, ∠6 and
∠7, ∠7 and ∠8
c. Sample answer: Twelve of the 28 angle pairs are
congruent. So, the likelihood of selecting a pair of
congruent angles is .
ANSWER:
a. Sample answer: There are 28 possible angle
pairings. The first angle can be paired with seven
others, then the second angle can be paired with six
others since it has already been paired with the first
angle. The number of pairings is the sum of the
number of angles each subsequent angle can be
paired with, 7 + 6 + 5 + 4 + 3 + 2 + 1 or 28 pairings.
b. Sample answer: There are two possible
relationships between the pairs of angles. Two angles
chosen will be either congruent or supplementary.
c. Sample answer: Twelve of the 28 angle pairs are
congruent. So, the likelihood of selecting a pair of
congruent angles is .
41.MULTIPLE REPRESENTATIONS In this
problem, you will investigate the relationship between
same-side exterior angles.
a. GEOMETRY Draw five pairs of parallel lines, m
and n, a and b, r and s, j and k, and x and y, cut by a
transversal t, and measure the four angles on one
side of t.
b. TABULAR Record your data in a table.
c. VERBAL Make a conjecture about the
relationship between the pair of angles formed on the
exterior of parallel lines and on the same side of the
transversal.
d. LOGICAL What type of reasoning did you use to
form your conjecture? Explain.
e. PROOF Write a proof of your conjecture.
SOLUTION:
a. Sample answer for m and n:
b. Sample answer:
c. Sample answer: In the diagram, ∠1 and ∠4 are a
pair of exterior angles on the same side of the
transversal.Thesumofm∠1 and m∠4 for each
row is 60 + 120 = 180, 45 + 135 = 180, 70 + 110 =
180, 90 + 90 = 180, and 25 + 155 = 180. A pair of
angles whose sum is 180 are supplementary angles.
Therefore, angles on the exterior of a pair of parallel
lines located on the same side of the transversal are
supplementary.
d. Inductive; a pattern was used to make a
conjecture.
e. Given: parallel lines m and n cut by transversal t.
Prove: 1and 4aresupplementary.
Proof:
1. Lines m and n are parallel and cut by transversal t.
(Given)
2. (Suppl.Thm.)
3. (Corr.anglesare .)
4. (Def.ofcongruence.)
5. (Subs.)
6. Angle 1 and angle 4 are supplementary. (Def. of
supplementary angles.)
ANSWER:
a. Sample answer for m and n:
b. Sample answer:
c. Sample answer: Angles on the exterior of a pair of
parallel lines located on the same side of the
transversal are supplementary.
d. Inductive; a pattern was used to make a
conjecture.
e. Given: parallel lines m and n cut by transversal t
Prove: 1 and 4 are supplementary.
Proof:
1. Lines m and n are parallel and cut by transversal t.
(Given)
2. m1 + m2 = 180 (Suppl. Thm.)
3. 2 4 (Corr. s are .)
4. m2 = m4 (Def. of congruence.)
5. m1 + m4 = 180 (Subs.)
6. 1 and 4 are supplementary. (Def. of
supplementary s.)
42.WRITING IN MATH If line a is parallel to line b
and describetherelationshipbetweenlines
b and c. Explain your reasoning.
SOLUTION:
Lines b and c are perpendicular. Since and
formalinearpair, .
so . Substituting, , so
and . So, lines a and c are
perpendicular. By Theorem 3.4, since transversal c is
perpendicular to line a and lines a and b are parallel,
then line c is perpendicular to line b.
ANSWER:
Lines b and c are perpendicular. Since 1 and 2
form a linear pair, m1 + m2 = 180. 1 2,
so m1 = m2. Substituting, m1 + m1 = 180,
so m1 = 90 and m2 = 90. So, lines a and c are
perpendicular. By Theorem 3.4, since transversal c is
perpendicular to line a and lines a and b are parallel,
then line c is perpendicular to line b.
43.WRITING IN MATH Compare and contrast the
Alternate Interior Angles Theorem and the
Consecutive Interior Angles Theorem.
SOLUTION:
In both theorems, a pair of angles is formed when
two parallel lines are cut by a transversal. However,
in the Alternate Interior Angles Theorem, each pair
of alternate interior angles that is formed are
congruent, whereas in the Consecutive Interior
Angles Theorem, each pair of angles formed is
supplementary.
ANSWER:
In both theorems, a pair of angles is formed when
two parallel lines are cut by a transversal. However,
in the Alternate Interior Angles Theorem, each pair
of alternate interior angles that is formed are
congruent, whereas in the Consecutive Interior
Angles Theorem, each pair of angles formed is
supplementary.
44.OPEN ENDED Draw a pair of parallel lines cut by
a transversal and measure the two exterior angles on
the same side of the transversal. Include the
measures on your drawing. Based on the pattern you
have seen for naming other pairs of angles, what do
you think the name of the pair you measured would
be?
SOLUTION:
Consecutive Exterior Angles or Same-Side Exterior
Angles
ANSWER:
Consecutive Exterior Angles or Same-Side Exterior
Angles
45.CHALLENGE Find x and y.
SOLUTION:
To find x and y, we will write two equation and solve
the system. In the figure, we are given a pair of
consecutive interior angles [xº and y2º], alternate
interior angles ([y2º and (8y – 15)º ]and
supplementary angles [xº and (8y – 15)º]. Use the
supplementary angles and consecutive interior angles
sincetheyarebothequationto180.
Supplementaryanglesequation:
Consecutive Interior angles equation:
Name the equation.
Substitute inEquation2.
Substitute inEquation2.
Thus, or .
ANSWER:
x = 171 or x = 155;
y = 3 or y = 5
46.REASONING Determine the minimum number of
angle measures you would have to know to find the
measures of all the angles formed by two parallel
lines cut by a transversal. Explain.
SOLUTION:
One;
Sample answer: Once the measure of one angle is
known, the rest of the angles are either congruent or
supplementary to the given angle.
ANSWER:
One; sample answer: Once the measure of one angle
is known, the rest of the angles are either congruent
or supplementary to the given angle.
47.Suppose 4 and 5 form a linear pair. If m1 =
2x, m2 = 3x – 20, and m3 = x – 4, what is m
3?
A 26°
B 28°
C 30°
D 32°
SOLUTION:
Use the definition of supplementary angles to find x.
To find , substitute in .
So, the correct option is C.
ANSWER:
C
48.SAT/ACT A farmer raises chickens and pigs. If his
animals have a total of 120 heads and a total of 300
feet, how many chickens does the farmer have?
F 60
G 70
H 80
J 90
SOLUTION:
Chickens and pigs have one head each and also they
have 2 feet and 4 feet respectively. Let x be the
number of pigs and y be the number chickens.
Solve the first equation for x.
Substitute 120 – y for x in equation (2).
Therefore, the farmer has 90 chickens. So, the
correct option is J.
ANSWER:
J
49.SHORT RESPONSE If thenwhichofthe
following statements must be true?
I. 3 and 6 are Alternate Interior Angles.
II. 4 and 6 are Consecutive Interior Angles.
III. 1 and 7 are Alternate Exterior Angles.
SOLUTION:
In the figure, angles 3 and 6 are alternate interior
angles, angles 4 and 6 consecutive interior angles,
and angles 1 and 7 are consecutive exterior angles.
So, the statements I and II are true.
ANSWER:
I and II
50.ALGEBRA If –2 + x = –6, then –17 – x =
A –13
B –4
C 13
D 21
SOLUTION:
Solve for x.
Substitute in
The correct choice is A.
ANSWER:
A
51.AVIATION Airplanes are assigned an altitude level
based on the direction they are flying. If one airplane
is flying northwest at 34,000 feet and another
airplane is flying east at 25,000 feet, describe the
type of lines formed by the paths of the airplanes.
Explain your reasoning.
SOLUTION:
Since the planes are flying at different altitude levels,
they are flying in different planes. The lines formed
by the path of the planes will not intersect since they
are flying in different directions. Lines that are not
coplanar and do not intersect are skew lines.
Therefore, the lines formed by the paths of the
airplanes are skew lines.
ANSWER:
Skew lines; the planes are flying in different
directions and at different altitudes.
Use the given information to find the measure
of each numbered angle.
52.If 1 and 2 form a linear pair and m2 = 67.
SOLUTION:
Since the angles 1 and 2 are linear pairs, they are
supplementary.
Substitute.
ANSWER:
m1 = 113
53. 6 and 8 are complementary; m8 = 47.
SOLUTION:
Angles 6 and 8 are complementary.
In the figure, angles 6, 7 and 8 form a linear pair
.
ANSWER:
m6 = 43, m7 = 90
54.m4 = 32
SOLUTION:
We know that vertical angles are congruent.
So,
Substitute.
In the figure,
So,
ANSWER:
m3 = 90, m5 = 58
55.TRAINS A train company wants to provide routes
to New York City, Dallas, Chicago, Los Angeles,
San Francisco, and Washington, D.C. An engineer
draws lines between each pair of cities on a map. No
three of the cities are collinear. How many lines did
the engineer draw?
SOLUTION:
The engineer drew 15 lines.
ANSWER:
15
Simplify each expression.
56.
SOLUTION:
ANSWER:
57.
SOLUTION:
ANSWER:
58.
SOLUTION:
ANSWER:
59.
SOLUTION:
ANSWER:
1
60.
SOLUTION:
ANSWER:
61.
SOLUTION:
ANSWER:
In the figure, m1 = 94. Find the measure of
each angle. Tell which postulate(s) or theorem
(s) you used.
1. 3
SOLUTION:
In the figure, angles 1 and 3 are corresponding
angles. Use the Corresponding Angles Postulate.
If two parallel lines are cut by a transversal, then
each pair of corresponding angles are congruent.
ANSWER:
94; Corresponding Angle Postulate
2. 5
SOLUTION:
In the figure, angles 1 and 5 are alternate exterior
angles.
Use the Alternate Exterior Angles Theorem.
If two parallel lines are cut by a transversal, then
each pair of alternate interior angles is congruent.
ANSWER:
94; Alt. Ext. s Thm.
3. 4
SOLUTION:
In the figure, angles 1 and 3 are corresponding
angles. Use the Corresponding Angles Postulate:
If two parallel lines are cut by a transversal, then
each pair of corresponding angles is congruent.
ANSWER:
86; Corresponding Angle Postulate and Supplement
Angle Thm.
In the figure, m4 = 101. Find the measure of
each angle. Tell which postulate(s) or theorem
(s) you used.
4. 6
SOLUTION:
In the figure, angles 4 and 6 are alternate interior
angles.
Use the Alternate interior Angles Theorem.
If two parallel lines are cut by a transversal, then
each pair of alternate interior angles is congruent.
ANSWER:
101; Alt. Int. s Thm.
5. 7
SOLUTION:
In the figure, angles 4 and 5 are consecutive interior
angles.
In the figure, angles 5 and 7 are vertical angles.
ANSWER:
79; Vertical Angle Thm. Cons. Int. s Thm.
6. 5
SOLUTION:
In the figure, angles 4 and 5 are consecutive interior
angles.
ANSWER:
79; Cons. Int. s Thm.
7.ROADS In the diagram, the guard rail is parallel to
the surface of the roadway and the vertical supports
are parallel to each other. Find the measures of
angles 2, 3, and 4.
SOLUTION:
Use the Alternate Interior Angles Theorem,
Definition of Supplementary Angles and
Corresponding Angles Postulate to find m 4.
So,m 4 = 87.
ANSWER:
m2 = 93, m3 = 87, m4 = 87
Find the value of the variable(s) in each figure.
Explain your reasoning.
8.
SOLUTION:
Use the definition of supplementary angles to find
. Then use the Alternate Interior Angles
Theorem to find .
ANSWER:
x = 125 by the Supplement Thm.;
y = 125 by the Alt. Int. s Thm.
9.
SOLUTION:
Use the Alternate Exterior Angles Theorem to find
x.
ANSWER:
x = 114 by the Alt. Ext. s Thm.
10.
SOLUTION:
Use the Alternate Interior Angles Theorem to find x.
ANSWER:
x = 70 by the Alt. Int. s Thm.
In the figure, m11 = 62 and m14 = 38. Find
the measure of each angle. Tell which postulate
(s) or theorem(s) you used.
11. 4
SOLUTION:
In the figure, angles 4 and 11 are corresponding
angles.
ANSWER:
62; Corr. s Post.
12. 3
SOLUTION:
In the figure, angles 4 and 11 are corresponding
angles and angles 3 and 4 are vertical angles.
ANSWER:
62; Corresponding s Post. and Vertical Thm.or
Alt. Ext. Thm.
13. 12
SOLUTION:
In the figure, angles 12 and 11 are supplementary
angles.
ANSWER:
118; Def. Supp. s
14. 8
SOLUTION:
In the figure, angles 8 and 11 are vertical angles.
ANSWER:
62; Vertical Angle Thm.
15. 6
SOLUTION:
In the figure, angles 14 and 6 are corresponding
angles.
ANSWER:
38; Corr. s Post.
16. 2
SOLUTION:
The angles 1 and 14 are alternate exterior
angles and so are congruent. and angles 3 and
11 are alternate exterior angles and so are
congruent. By Supplementary Theorem, m1 + m
2 + m3 = 180.
ANSWER:
80; Alt. Ext. s Post. and Supp. Thm.
17. 10
SOLUTION:
In the figure, angles 14 and 10 are supplementary
angles.
ANSWER:
142; Supplement Angles Thm.
18. 5
SOLUTION:
Use definition of supplementary angles,
Corresponding Angles Postulate and the Alternate
Interior Angles Theorem .
ANSWER:
80; Vertical Angles Thm.
19. 1
SOLUTION:
In the figure, angles 1 and 14 are alternate exterior
angles.
ANSWER:
38; Alt. Ext. s Thm.
CCSS MODELING A solar dish collects energy
by directing radiation from the Sun to a receiver
located at the focal point of the dish. Assume
that the radiation rays are parallel. Determine
the relationship between each pair of angles and
explain your reasoning.
Refer to Page 183.
20. 1 and 2
SOLUTION:
If the radiation rays form parallel lines, then ∠1 and
∠2 are consecutive interior angles. So, according to
the Consecutive Interior Angles Theorem, ∠1 and
∠2 are supplementary.
ANSWER:
supplementary; Consecutive Interior Angles
21. 1 and 3
SOLUTION:
If the radiation rays form parallel lines, then ∠1 and
∠3 are corresponding angles. So, according to the
Corresponding Angles Postulate, ∠1 and ∠3 are
congruent.
ANSWER:
congruent; Corresponding Angles
22. 4 and 5
SOLUTION:
If the radiation rays form parallel lines, then ∠4 and
∠5 are alternate exterior angles. So, according to the
Alternate Exterior Angles Theorem, ∠4 and ∠5 are
congruent.
ANSWER:
congruent; Alternate Exterior Angles
23. 3 and 4
SOLUTION:
If the radiation rays form parallel lines, then ∠3 and
∠5 are a linear pair of angles.. So, according to the
definition of linear pairs, ∠3 and ∠5 are
supplementary.∠4 and ∠5 are alternate exterior
angles. So, by the Alternate Exterior Angles
Theorem, ∠4 is congruent to ∠5. Then,
Therefore, ∠3 is supplementary to ∠4 by the
definition of supplementary angles.
ANSWER:
supplementary; since 3 and 5 are a linear pair,
they are supplementary. 4 and 5 are congruent
because they are alternate exterior angles, so 3 is
supplementary to 4.
Find the value of the variable(s) in each figure.
Explain your reasoning.
24.
SOLUTION:
Use Corresponding Angles Postulate and definition
of supplementary angles to find x.
ANSWER:
y = 114 by the Corresponding Angles Postulate; x =
54 by the Supplement Theorem
25.
SOLUTION:
Use the Corresponding Angles Postulate and
Supplement Theorem to find x and y.
ANSWER:
x = 40 by the Corresponding Angles Postulate; y =
50 by the Supplement Theorem
26.
SOLUTION:
Use the Vertical Angle Theorem and Consecutive
Interior Angles Theorem to find x.
ANSWER:
x = 63 by the Vertical Angle Theorem and the
Consecutive Interior Angles Theorem
27.
SOLUTION:
Use the Consecutive Interior Angles Theorem to find
x and y.
ANSWER:
x = 42 by the Consecutive Interior Angles Theorem;
y = 14 by the Consecutive Interior Angles Theorem
28.
SOLUTION:
Use the Alternate Interior Angles Theorem and
Consecutive Interior Angles Theorem to find x and y.
ANSWER:
x = 54 by the Alternate Interior Angles Theorem; y =
12 by the Consecutive Interior Angles Theorem
29.
SOLUTION:
Use the Consecutive Interior Angles Theorem
anddefinitionofsupplementaryanglestofind x and
y.
ANSWER:
x = 60 by the Consecutive Interior Angles Theorem;
y = 10 by the Supplement Theorem
30.PROOF Copy and complete the proof of Theorem
3.2.
Given: isatransversal.
Prove: 1 and 2 are supplementary;
3 and 4 are supplementary.
Proof:
SOLUTION:
ANSWER:
STORAGE When industrial shelving needs to
be accessible from either side, additional
support is provided on the side by transverse
members. Determine the relationship between
each pair of angles and explain your reasoning.
31. 1 and 8
SOLUTION:
1 and 8 are Alternate interior angles. Therefore
1 and 8 are congruent.
ANSWER:
Congruent; Alternate interior angles
32. 1 and 5
SOLUTION:
1 and 5 are Corresponding angles. Therefore,
they are congruent.
ANSWER:
Congruent; Corresponding angles
33. 3 and 6
SOLUTION:
3 and 6 are Vertical angles. Therefore Vertical
angles are congruent.
ANSWER:
Congruent; Vertical angles are congruent
34. 1 and 2
SOLUTION:
All vertical and horizontal lines are perpendicular at
their point of intersection.By definition of
perpendicular,theyformrightangles.∠1 and ∠2
are adjacent angles. By the Angle Addition Postulate,
m∠1+m∠2 = 90. Since the sum of the two angles is
90, ∠1 and ∠2 are complementary angles.
ANSWER:
Complementary; because the vertical and horizontal
lines are perpendicular, they form right angles.
35.CCSS ARGUMENTS Write a two-column proof of
the Alternate Exterior Angles Theorem.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. (Corr. s Post.)
3. (Vertical s Thm.)
4. (Trans.Prop.)
ANSWER:
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. 1 5, 2 6 (Corr. s Post.)
3. 5 8, 6 7 (Vertical s Thm.)
4. 1 8, 2 7 (Trans. Prop.)
36.BRIDGES Refer to the diagram of the double
decker Michigan Avenue Bridge in Chicago, Illinois.
The two levels of the bridge, and its diagonal braces,
are parallel.
a. How are the measures of the odd-numbered
angles related? Explain.
b. How are the measures of the even-numbered
angles related? Explain.
c. How are any pair of angles in which one is odd
and the other is even related?
d. What geometric term(s) can be used to relate the
two roadways contained by the bridge?
SOLUTION:
a. The top and bottom levels of the bridge are
parallel, so the lines formed by the edge of each level
are parallel, and by using the diagonal braces as
transversals and the Alternate Interior Angles
Theorem ∠1 ∠3, ∠5 ∠7, ∠9 ∠11, and
∠13 ∠15.
The diagonal braces are parallel, so by using the
vertical braces as transversals and the Alternate
Interior Angles Theorem ∠4 ∠6, ∠8 ∠10,
and ∠12 ∠14.
Since the vertical braces are perpendicular to the
levels of the bridge, ∠3 and ∠4, ∠5 and ∠6, ∠7 and
∠8, ∠9 and ∠10, ∠11 and ∠12, and ∠13 and ∠14
arepairsofcomplementaryangles.
By the Congruent Complements Theorem, ∠3
∠5, ∠7 ∠9, and ∠11 ∠13. So, ∠1
∠3 ∠5 ∠7 ∠9 ∠11 ∠13 ∠13
by the Transitive Property of Congruence. So, all of
the odd numbered angles are alternate interior angles
related by the diagonal transversals or are
complements of even numbered alternate interior
angles related by the vertical transversals. Therefore,
they are all congruent.
b. All the vertical braces are parallel since all vertical
lines are parallel. Using the diagonal braces as
transversals to the vertical braces and the Alternate
Interior Angles Theorem, ∠2 ∠4, ∠6 ∠8,
∠10 ∠12, and ∠14 ∠16.
Using the vertical braces as transversals between the
diagonal braces and the Alternate Interior Angles
Theorem, ∠4 ∠6, ∠8 ∠10, and ∠12
∠14. So, ∠2 ∠4 ∠6 ∠8 ∠10
∠12 ∠14 ∠16 by the Transitive Property of
Congruence.
All of the even numbered angles are alternate interior
angles related by either the diagonal transversals or
the vertical transversals. Therefore, they are all
congruent.
c. Complementary; since the vertical supports and
the horizontal supports are perpendicular, angle pairs
like angle 1 and angle 2 must be complementary.
Since all of the odd numbered angles are congruent
and all of the even numbered angles are congruent,
any pair of angles that has one odd and one even
number will be complementary.
d. Since the two levels (or surfaces) of the bridge
are parallel, the geometric term that best represents
the two roadways contained by the bridge is parallel
planes.
ANSWER:
a. Congruent; all of the odd numbered angles are
alternateinterioranglesrelatedbythediagonal
transversals or are complements of even numbered
alternate interior angles related by the vertical
transversals, so they are all congruent.
b. Congruent; all of the even numbered angles are
alternate interior angles related by either the diagonal
transversals or the vertical transversals, so they are
all congruent.
c. Complementary; since the vertical supports and
the horizontal supports are perpendicular, angle pairs
like 1 and 2 must be complementary. Since all
of the odd numbered angles are congruent and all of
the even numbered angles are congruent, any pair of
angles that has one odd and one even number will be
complementary.
d. parallel planes
37.PROOF In a plane, prove that if a line is
perpendicular to one of two parallel lines, then it is
perpendicular to the other.
SOLUTION:
Given:
Prove:
Proof:
Statements (Reasons)
1. (Given)
2. Angle 1 is a right angle. (Def. of )
3. (Def.ofrt. )
4. (Corr. Post.)
5. (Def.of )
6. (Subs.)
7. isarightangle.(Def.ofrt. )
8. (Def.of lines)
ANSWER:
Given:
Prove:
Proof:
Statements (Reasons)
1. m || n, t ⊥ m (Given)
2. 1 is a right angle. (Def. of ⊥)
3. m1 = 90 (Def. of rt. s)
4. 1 2 (Corr. s Post.)
5. m1 = m2 (Def. of s)
6. m2 = 90 (Subs.)
7. 2 is a right angle. (Def. of rt. s)
8. t ⊥ n (Def. of ⊥ lines)
CCSSTOOLSFindx. (Hint: Draw an auxiliary
line.)
38.
SOLUTION:
Draw an auxiliary line to construct a triangle.
Then label the angles a°,b°,andc°.Byfindingthe
measures for angles a and b, we can use the
Triangle Angle Sum theorem to find angle c. Angles
c and xareverticalangles.
Use the definition of supplementary angles to find a.
Find angle b.
Find angle c.
Find angle x.
So, x = 22.
ANSWER:
22
39.
SOLUTION:
Draw an auxiliary line to construct a triangle.By
creating a triangle, we can sue the Triangle
Angle Sum Theorem and definition of
supplementary angles to find x. Label the
angles.
First find angle a.
Find angle b.
Find angle c.
Find angle d.
Find angle x
.
So x = 130º.
ANSWER:
130
40.PROBABILITY Suppose you were to pick any two
angles in the figure below.
a. How many possible angle pairings are there?
Explain.
b. Describe the possible relationships between the
measures of the angles in each pair. Explain.
c. Describe the likelihood of randomly selecting a
pair of congruent angles. Explain your reasoning.
SOLUTION:
a. Sample answer: There are 28 possible angle
pairings. The first angle can be paired with seven
others, then the second angle can be paired with six
others since it has already been paired with the first
angle. The number of pairings is the sum of the
number of angles each subsequent angle can be
paired with, 7 + 6 + 5 + 4 + 3 + 2 + 1 or 28 pairings.
b. Sample answer: Because the two lines being
transversed are parallel, there are only two possible
relationships between the pairs of angles. Each pair
of angles chosen will be either congruent or
supplementary.
Congruent pairs: ∠1 and ∠3, ∠1 and ∠5, ∠1 and
∠7, ∠3 and ∠5, ∠3 and ∠7, ∠5 and ∠7, ∠2 and
∠4, ∠2 and ∠6, ∠2 and ∠8, ∠4 and ∠6, ∠4 and
∠8, ∠6 and ∠8
Supplementary pairs: ∠1 and ∠2, ∠1 and ∠4, ∠1
and ∠6, ∠1 and ∠8, ∠2 and ∠3, ∠2 and ∠5, ∠2
and ∠7, ∠3 and ∠4, ∠3 and 6, ∠3 and ∠8, ∠4 and
∠5, ∠4 and ∠7, ∠5 and ∠6, ∠5 and ∠8, ∠6 and
∠7, ∠7 and ∠8
c. Sample answer: Twelve of the 28 angle pairs are
congruent. So, the likelihood of selecting a pair of
congruent angles is .
ANSWER:
a. Sample answer: There are 28 possible angle
pairings. The first angle can be paired with seven
others, then the second angle can be paired with six
others since it has already been paired with the first
angle. The number of pairings is the sum of the
number of angles each subsequent angle can be
paired with, 7 + 6 + 5 + 4 + 3 + 2 + 1 or 28 pairings.
b. Sample answer: There are two possible
relationships between the pairs of angles. Two angles
chosen will be either congruent or supplementary.
c. Sample answer: Twelve of the 28 angle pairs are
congruent. So, the likelihood of selecting a pair of
congruent angles is .
41.MULTIPLE REPRESENTATIONS In this
problem, you will investigate the relationship between
same-side exterior angles.
a. GEOMETRY Draw five pairs of parallel lines, m
and n, a and b, r and s, j and k, and x and y, cut by a
transversal t, and measure the four angles on one
side of t.
b. TABULAR Record your data in a table.
c. VERBAL Make a conjecture about the
relationship between the pair of angles formed on the
exterior of parallel lines and on the same side of the
transversal.
d. LOGICAL What type of reasoning did you use to
form your conjecture? Explain.
e. PROOF Write a proof of your conjecture.
SOLUTION:
a. Sample answer for m and n:
b. Sample answer:
c. Sample answer: In the diagram, ∠1 and ∠4 are a
pair of exterior angles on the same side of the
transversal.Thesumofm∠1 and m∠4 for each
row is 60 + 120 = 180, 45 + 135 = 180, 70 + 110 =
180, 90 + 90 = 180, and 25 + 155 = 180. A pair of
angles whose sum is 180 are supplementary angles.
Therefore, angles on the exterior of a pair of parallel
lines located on the same side of the transversal are
supplementary.
d. Inductive; a pattern was used to make a
conjecture.
e. Given: parallel lines m and n cut by transversal t.
Prove: 1and 4aresupplementary.
Proof:
1. Lines m and n are parallel and cut by transversal t.
(Given)
2. (Suppl.Thm.)
3. (Corr.anglesare .)
4. (Def.ofcongruence.)
5. (Subs.)
6. Angle 1 and angle 4 are supplementary. (Def. of
supplementary angles.)
ANSWER:
a. Sample answer for m and n:
b. Sample answer:
c. Sample answer: Angles on the exterior of a pair of
parallel lines located on the same side of the
transversal are supplementary.
d. Inductive; a pattern was used to make a
conjecture.
e. Given: parallel lines m and n cut by transversal t
Prove: 1 and 4 are supplementary.
Proof:
1. Lines m and n are parallel and cut by transversal t.
(Given)
2. m1 + m2 = 180 (Suppl. Thm.)
3. 2 4 (Corr. s are .)
4. m2 = m4 (Def. of congruence.)
5. m1 + m4 = 180 (Subs.)
6. 1 and 4 are supplementary. (Def. of
supplementary s.)
42.WRITING IN MATH If line a is parallel to line b
and describetherelationshipbetweenlines
b and c. Explain your reasoning.
SOLUTION:
Lines b and c are perpendicular. Since and
formalinearpair, .
so . Substituting, , so
and . So, lines a and c are
perpendicular. By Theorem 3.4, since transversal c is
perpendicular to line a and lines a and b are parallel,
then line c is perpendicular to line b.
ANSWER:
Lines b and c are perpendicular. Since 1 and 2
form a linear pair, m1 + m2 = 180. 1 2,
so m1 = m2. Substituting, m1 + m1 = 180,
so m1 = 90 and m2 = 90. So, lines a and c are
perpendicular. By Theorem 3.4, since transversal c is
perpendicular to line a and lines a and b are parallel,
then line c is perpendicular to line b.
43.WRITING IN MATH Compare and contrast the
Alternate Interior Angles Theorem and the
Consecutive Interior Angles Theorem.
SOLUTION:
In both theorems, a pair of angles is formed when
two parallel lines are cut by a transversal. However,
in the Alternate Interior Angles Theorem, each pair
of alternate interior angles that is formed are
congruent, whereas in the Consecutive Interior
Angles Theorem, each pair of angles formed is
supplementary.
ANSWER:
In both theorems, a pair of angles is formed when
two parallel lines are cut by a transversal. However,
in the Alternate Interior Angles Theorem, each pair
of alternate interior angles that is formed are
congruent, whereas in the Consecutive Interior
Angles Theorem, each pair of angles formed is
supplementary.
44.OPEN ENDED Draw a pair of parallel lines cut by
a transversal and measure the two exterior angles on
the same side of the transversal. Include the
measures on your drawing. Based on the pattern you
have seen for naming other pairs of angles, what do
you think the name of the pair you measured would
be?
SOLUTION:
Consecutive Exterior Angles or Same-Side Exterior
Angles
ANSWER:
Consecutive Exterior Angles or Same-Side Exterior
Angles
45.CHALLENGE Find x and y.
SOLUTION:
To find x and y, we will write two equation and solve
the system. In the figure, we are given a pair of
consecutive interior angles [xº and y2º], alternate
interior angles ([y2º and (8y – 15)º ]and
supplementary angles [xº and (8y – 15)º]. Use the
supplementary angles and consecutive interior angles
sincetheyarebothequationto180.
Supplementaryanglesequation:
Consecutive Interior angles equation:
Name the equation.
Substitute inEquation2.
Substitute inEquation2.
Thus, or .
ANSWER:
x = 171 or x = 155;
y = 3 or y = 5
46.REASONING Determine the minimum number of
angle measures you would have to know to find the
measures of all the angles formed by two parallel
lines cut by a transversal. Explain.
SOLUTION:
One;
Sample answer: Once the measure of one angle is
known, the rest of the angles are either congruent or
supplementary to the given angle.
ANSWER:
One; sample answer: Once the measure of one angle
is known, the rest of the angles are either congruent
or supplementary to the given angle.
47.Suppose 4 and 5 form a linear pair. If m1 =
2x, m2 = 3x – 20, and m3 = x – 4, what is m
3?
A 26°
B 28°
C 30°
D 32°
SOLUTION:
Use the definition of supplementary angles to find x.
To find , substitute in .
So, the correct option is C.
ANSWER:
C
48.SAT/ACT A farmer raises chickens and pigs. If his
animals have a total of 120 heads and a total of 300
feet, how many chickens does the farmer have?
F 60
G 70
H 80
J 90
SOLUTION:
Chickens and pigs have one head each and also they
have 2 feet and 4 feet respectively. Let x be the
number of pigs and y be the number chickens.
Solve the first equation for x.
Substitute 120 – y for x in equation (2).
Therefore, the farmer has 90 chickens. So, the
correct option is J.
ANSWER:
J
49.SHORT RESPONSE If thenwhichofthe
following statements must be true?
I. 3 and 6 are Alternate Interior Angles.
II. 4 and 6 are Consecutive Interior Angles.
III. 1 and 7 are Alternate Exterior Angles.
SOLUTION:
In the figure, angles 3 and 6 are alternate interior
angles, angles 4 and 6 consecutive interior angles,
and angles 1 and 7 are consecutive exterior angles.
So, the statements I and II are true.
ANSWER:
I and II
50.ALGEBRA If –2 + x = –6, then –17 – x =
A –13
B –4
C 13
D 21
SOLUTION:
Solve for x.
Substitute in
The correct choice is A.
ANSWER:
A
51.AVIATION Airplanes are assigned an altitude level
based on the direction they are flying. If one airplane
is flying northwest at 34,000 feet and another
airplane is flying east at 25,000 feet, describe the
type of lines formed by the paths of the airplanes.
Explain your reasoning.
SOLUTION:
Since the planes are flying at different altitude levels,
they are flying in different planes. The lines formed
by the path of the planes will not intersect since they
are flying in different directions. Lines that are not
coplanar and do not intersect are skew lines.
Therefore, the lines formed by the paths of the
airplanes are skew lines.
ANSWER:
Skew lines; the planes are flying in different
directions and at different altitudes.
Use the given information to find the measure
of each numbered angle.
52.If 1 and 2 form a linear pair and m2 = 67.
SOLUTION:
Since the angles 1 and 2 are linear pairs, they are
supplementary.
Substitute.
ANSWER:
m1 = 113
53. 6 and 8 are complementary; m8 = 47.
SOLUTION:
Angles 6 and 8 are complementary.
In the figure, angles 6, 7 and 8 form a linear pair
.
ANSWER:
m6 = 43, m7 = 90
54.m4 = 32
SOLUTION:
We know that vertical angles are congruent.
So,
Substitute.
In the figure,
So,
ANSWER:
m3 = 90, m5 = 58
55.TRAINS A train company wants to provide routes
to New York City, Dallas, Chicago, Los Angeles,
San Francisco, and Washington, D.C. An engineer
draws lines between each pair of cities on a map. No
three of the cities are collinear. How many lines did
the engineer draw?
SOLUTION:
The engineer drew 15 lines.
ANSWER:
15
Simplify each expression.
56.
SOLUTION:
ANSWER:
57.
SOLUTION:
ANSWER:
58.
SOLUTION:
ANSWER:
59.
SOLUTION:
ANSWER:
1
60.
SOLUTION:
ANSWER:
61.
SOLUTION:
ANSWER:
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3-2 Angles and Parallel Lines
