Find the measures of each numbered angle.
1.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. Let x be the measure of unknown angle in the
figure.
2.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
In the figure,
In the figure, andtheanglemeasuring39°are
congruent.
So,
Find each measure.
3.m2
SOLUTION:
BytheExteriorAngleTheorem, .
4.mMPQ
SOLUTION:
By the Exterior Angle Theorem,
DECK CHAIRS The brace of this deck chair
forms a triangle with the rest of the chair’s
frame as shown. If m1 = 102 and m3 = 53,
find each measure.
Refer to the figure on page 250.
5.m4
SOLUTION:
By the Exterior Angle Theorem,
Substitute.
6.m6
SOLUTION:
In the figure, and form a linear pair. So,
7.m2
SOLUTION:
By the Exterior Angle Theorem,
Substitute.
The sum of the measures of the angles of a triangle
is 180.
So,
Substitute.
8.m5
SOLUTION:
Angles 4 and 5 form a linear pair. Use the Exterior
Angle Theorem to find first and then use the
fact that the sum of the measures of the two angles
of a linear pair is 180.
By the Exterior Angle Theorem,
Substitute.
In the figure, and form a linear pair. So,
CCSSREGULARITYFind each measure.
9.m1
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
10.m3
SOLUTION:
In the figure, and angleformalinearpair.
So,
The sum of the measures of the angles of a triangle
is 180. So, .
Substitute.
11.m2
SOLUTION:
In the figure, and angleformalinearpair.
So,
Find the measure of each numbered angle.
12.Refer to the figure on page 250.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
13.Refer to the figure on page 250.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
14.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.In .
Here, and arecongruentangles.Bythe
definition of congruence,
In .
Since , .
15.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.In .
In the figure, and are vertical angles. Since
vertical angles are congruent, .
In .
Substitute.
16.AIRPLANES The path of an airplane can be
modeled using two sides of a triangle as shown. The
distance covered during the plane’s ascent is equal to
the distance covered during its descent.
a. Classify the model using its sides and angles.
b. The angles of ascent and descent are congruent.
Find their measures.
SOLUTION:
a. The triangle has two congruent sides. So, it is
isosceles. One angle of the triangle measures 173, so
it is a obtuse angle. Since the triangle has an obtuse
angle, it is an obtuse triangle.
b. Let x be the angle measure of ascent and descent.
We know that
the sum of the measures of the angles of a triangle is
180. So, .
The angle of ascent is 3.5 and the angle of descent
is 3.5.
Find each measure.
17.m1
SOLUTION:
By the Exterior Angle Theorem,
Find .
18.m3
SOLUTION:
By the Exterior Angle Theorem,
Find .
19.m2
SOLUTION:
By the Exterior Angle Theorem,
Solve for .
That is,
20.m4
SOLUTION:
By the Exterior Angle Theorem,
Solve for .
That is,
21.mABC
SOLUTION:
By the Exterior Angle Theorem,
Find .
That is, .
Substitute in
22.mJKL
SOLUTION:
By the Exterior Angle Theorem,
.
Find x.
That is, .
Substitute in
23.WHEELCHAIR RAMP Suppose the wheelchair
rampshownmakesa12°anglewiththeground.
What is the measure of the angle the ramp makes
with the van door?
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
Let x be the measure of the angle the ramp makes
with the van door.
CCSSREGULARITYFindeachmeasure.
24.m1
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
25.m2
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
26.m3
SOLUTION:
By the Exterior Angle Theorem, .
That is,
27.m4
SOLUTION:
In the figure,
The sum of the measures of the angles of a triangle
is180.
In the figure, .
Substitute.
28.m5
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
29.m6
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
In the figure,
Substitute.
ALGEBRA Find the value of x. Then find the
measure of each angle.
30.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
Solve for x.
Substitute ineachmeasure.
31.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
Solve for x.
Substitute in2x.
32.
SOLUTION:
By the Exterior Angle Theorem,
.
Solve for x.
Substitute in5x + 62.
Substitute in3x + 47.
33.GARDENING A landscaper is forming an isosceles
triangle in a flowerbed using chrysanthemums. She
wants m A to be three times the measure of B
and C. What should the measure of each angle
be?
SOLUTION:
and
The sum of the measures of the angles of a triangle
is 180. In the figure, .
Substitute.
Since , .
Substitute in .
PROOF Write the specified type of proof.
34.flow proof of Corollary 4.1
SOLUTION:
Given:
isarightangle.
Prove: and are complementary.
Proof:
35.paragraph proof of Corollary 4.2
SOLUTION:
Given:
is a right angle.
Prove: There can be at most one right angle in a
triangle.
Proof: In , isarightangle.
. so
.
If werearightangle,then . But that
is impossible, so there cannot be two right angles in a
triangle.
Given:
isobtuse.
Prove: There can be at most one obtuse angle in a
triangle.
Proof: In , isobtuse.So .
. It must be that
. So, and must be acute.
CCSSREGULARITYFind the measure of
each numbered angle.
36.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
In the figure, .
Solve for
By the Exterior Angle Theorem, .
Substitute.
Also, .
Substitute.
37.
SOLUTION:
Look for pairs of vertical angles first. Here,
angle and are vertical angles, since
angle and are vertical angles, they are
congruent. By the definition of congruence,
.
Look for linear pairs next. Angles 5 and 7 are a
linear pair. Since ,
or70.
Next, the Triangle Angle Sum theorem can be used
to find .
From the diagram, . By the Exterior Angle
Theorem, .
Since congruent angles have equal measure,
.
Using the Triangle Angle Sum Theorem we know
that .
Using the Triangle Angle Sum Theorem we know
that .
Using the Triangle Angle Sum Theorem we know
that .
38.ALGEBRA Classify the triangle shown by its
angles. Explain your reasoning.
SOLUTION:
Obtuse; the sum of the measures of the three angles
of a triangle is 180. So, (15x + 1) + (6x + 5) + (4x –
1) = 180 and x = 7. Substituting 7 into the expressions
for each angle, the angle measures are 106, 47, and
27. Since the triangle has an obtuse angle, it is
obtuse.
39.ALGEBRA The measure of the larger acute angle
in a right triangle is two degrees less than three times
the measure of the smaller acute angle. Find the
measure of each angle.
SOLUTION:
Let x and y be the measure of the larger and smaller
acute angles in a right triangle respectively. Given
that . The sum of the measures of the
angles of a triangle is 180.
So,
Substitute.
Substitute in .
Thus the measure of the larger acute angle is 67 and
the measure of the smaller acute angle is 23.
40.Determine whether the following statement is true or
false. If false, give a counterexample. If true, give an
argument to support your conclusion.
If the sum of two acute angles of a triangle is
greater than 90, then the triangle is acute.
SOLUTION:
True; sample answer: Since the sum of the two acute
angles is greater than 90, the measure of the third
angle is a number greater than 90 subtracted from
180, which must be less than 90. Therefore, the
triangle has three acute angles and is acute.
41.ALGEBRA In , m X = 157, m Y = y, and
m Z = z. Write an inequality to describe the
possible measures of Z. Explain your reasoning.
SOLUTION:
z < 23; Sample answer: Since the sum of the
measures of the angles of a triangle is 180 and
, , so
If was0,then
wouldequal23.Butsinceananglemusthave
a measure greater than 0, mustbelessthan
23, so z < 23.
42.CARS Refer to the photo on page 252.
a. Find m1 and m2.
b. If the support for the hood were shorter than the
one shown, how would m1 change? Explain.
c. If the support for the hood were shorter than the
one shown, how would m2 change? Explain.
SOLUTION:
a. By the Exterior Angle Theorem,
So, Inthefigure,
Solve for .
b. Sample answer: The measure of wouldget
larger if the support were shorter because the hood
would be closer to the leg of the triangle that is along
the fender of the car.
c. Sample answer: The measure of wouldget
smaller if the support were shorter because
wouldgetlargerandtheyarealinearpair.
PROOF Write the specified type of proof.
43.two-column proof
Given: RSTUV is a pentagon.
Prove: m S + mSTU + mTUV + m V +
mVRS = 540
SOLUTION:
Proof: Statements (Reasons)
1. RSTUV is a pentagon. (Given)
2. ;
;
( Sum Thm.)
3. (Add. Prop.)
4. ;
;
( Addition)
5. (Subst.)
44.flow proof
Given: 3 5
Prove: m1 + m2 = m6 + m7
SOLUTION:
Proof:
45.MULTIPLE REPRESENTATIONS In this
problem, you will explore the sum of the measures of
the exterior angles of a triangle.
a. GEOMETRIC Draw five different triangles,
extending the sides and labeling the angles as shown.
Be sure to include at least one obtuse, one right, and
one acute triangle.
b. TABULAR Measure the exterior angles of each
triangle. Record the measures for each triangle and
the sum of these measures in a table.
c. VERBAL Make a conjecture about the sum of
the exterior angles of a triangle. State your
conjecture using words.
d. ALGEBRAIC State the conjecture you wrote in
part c algebraically.
e. ANALYTICAL Write a paragraph proof of your
conjecture.
SOLUTION:
a. Sample answer:
b. Sample answer:
c. Sample answer:
The sum of the measures of the exterior angles of a
triangle is 360.
d. m1 + m2 + m3 = 360
e. The Exterior Angle Theorem tells us that m3 =
mBAC + mBCA,
m2 = mBAC + mCBA,
m1 = mCBA + mBCA.
Through substitution,
m1+m2 + m3 = mCBA + mBCA +
mBAC + mCBA + mBAC + mBCA. Which
can be simplified to m1+m2 + m3 = 2m
BAC + 2mBCA + 2mCBA.
The Distributive Property can be applied and gives
m1+m2 + m3 = 2(mBAC + mBCA +
mCBA). The Triangle Angle-Sum Theorem tells us
that
mBAC + mBCA + mCBA = 180. Through
substitution we have m1+m2 + m3 = 2(180)
= 360.
46.CCSS CRITIQUE Curtis measured and labeled the
angles of the triangle as shown. Arnoldo says that at
least one of his measures is incorrect. Explain in at
least two different ways how Arnoldo knows that
this is true.
SOLUTION:
Sample answer: Corollary 4.2 states that there can
be at most one right or obtuse angle in a triangle.
Since this triangle is labeled with two obtuse angle
measures, 93 and 130, at least one of these measures
must be incorrect. Since by the Triangle Angle Sum
Theorem the sum of the interior angles of the triangle
mustbe180and37+93+130≠180,atleastoneof
these measures must be incorrect.
47.WRITING IN MATH Explain how you would find
the missing measures in the figure shown.
SOLUTION:
The measure of is the supplement of the exterior
angle with measure 110, so or 70.
Because the angles with measures b and c are
congruent, b = c . Using the Exterior Angle
Theorem, b + c = 110. By substitution, b + b = 110,
so 2b = 110 and b = 55. Because b = c, c = 55.
48.OPEN ENDED Construct a right triangle and
measure one of the acute angles. Find the measure
of the second acute angle using calculation and
explain your method. Confirm your result using a
protractor.
SOLUTION:
Sample answer:
I found the measure of the second angle by
subtractingthefirstanglefrom90°sincetheacute
angles of a right triangle are complementary.
49.CHALLENGE Find the values of y and z in the
figure.
SOLUTION:
In the figure, because they
are a linear pair and
because of the External Angle Theorem.
Simplify the equations and name them.
Subtract the equation (2) from (1).
Substitute in(1).
50.REASONING If an exterior angle adjacent to A
is acute, is acute,right,obtuse,orcanits
classification not be determined? Explain your
reasoning.
SOLUTION:
Obtuse; since the exterior angle is acute, the sum of
the remote interior angles must be acute, which
means the third angle must be obtuse. Therefore, the
triangle must be obtuse. Also, since the exterior angle
forms a linear pair with A, A must be obtuse since
two acute angles cannot be a linear pair.
51.WRITING IN MATH Explain why a triangle
cannot have an obtuse, acute, and a right exterior
angle.
SOLUTION:
Sample answer: Since an exterior angle is acute, the
adjacent angle must be obtuse. Since another exterior
angle is right, the adjacent angle must be right. A
triangle cannot contain both a right and an obtuse
angle because it would be more than 180 degrees.
Therefore, a triangle cannot have an obtuse, acute,
and a right exterior angle.
52.PROBABILITY Mr. Glover owns a video store
and wants to survey his customers to find what type
of movies he should buy. Which of the following
options would be the best way for Mr. Glover to get
accurate survey results?
A surveying customers who come in from 9 p.m.
until 10 p.m.
B surveying customers who come in on the weekend
C surveying the male customers
D surveying at different times of the week and day
SOLUTION:
The most accurate survey would ask a random
sampling of customers. Choices A, B, and C each
survey a specific group of customers. Choice D is a
random sample of customers so it will give Mr.
Glover the most accurate result.
53.SHORT RESPONSE Two angles of a triangle
havemeasuresof35°and80°.Describethepossible
values of the exterior angle measures of the triangle.
SOLUTION:
Sample answer: Since the sum of the measures of
the angles of a triangle is 180, the measure of the
third angle is 180 – (35 + 80) or 60. To find the
measures of the exterior angles, subtract each angle
measure from 180. The values for the exterior angle
ofthetriangleare100°,115°,and145°.
54.ALGEBRA Which equation is equivalent to 7x – 3
(2 – 5x) = 8x?
F 2x – 6 = 8
G 22x – 6 = 8x
H –8x – 6 = 8x
J22x + 6 = 8x
SOLUTION:
7x – 3(2 – 5x) = 8xOriginal equation
7x – 6 + 15x = 8xDistributive Property
22x – 6 = 8xSimplify.
So, the correct option is G.
55.SAT/ACT Joey has 4 more video games than
Solana and half as many as Melissa. If together they
have 24 video games, how many does Melissa have?
A 7
B 9
C 12
D 13
E 14
SOLUTION:
Let j, s, and m be the number of video games with
Joey, Solana, and Melissa respectively. Given
that , , and .
Substitute in .
Substitute in .
So, Melissa has 14 video games. The correct option
is E.
Classify each triangle as acute, equiangular,
obtuse, or right.
56.
SOLUTION:
Since all the angles are congruent, it is equiangular.
57.
SOLUTION:
One angle of the triangle measures 150, so it is an
obtuse angle. Since the triangle has an obtuse angle,
it is an obtuse triangle.
58.
SOLUTION:
One angle of the triangle measures 90, so it is a right
angle. Since the triangle has a right angle, it is a right
triangle.
COORDINATE GEOMETRY Find the
distance from P to .
59.Line containspoints(0,–2) and (1, 3). Point P has
coordinates (–4, 4).
SOLUTION:
Find the equation of the line Substitutethevalues
in the slope formula.
Then write the equation of this line using the point (1,
3).
Therefore, the equation of the line l is
Write an equation of the line w perpendicular to
through (–4, 4). Sincetheslopeofline is 5, the
slope of a line w is . Write the equation of line w
through (–4, 4) with slope 1.
Therefore, the equation of the line w is
Solve the system of equations to determine the point
of intersection. The left sides of the equations are the
same. So, equate the right sides and solve for x.
Use the value of x to find the value of y.
So,thepointofintersectionis(1,3)
Use the Distance Formula to find the distance
between the points (–4, 4) and (1, 3) .
Therefore, the distance between the two lines
is
60.Line contains points (–3, 0) and (3, 0). Point P has
coordinates (4, 3).
SOLUTION:
Here, line ishorizontal;infactitisthex-axis. So, a
line perpendicular to isvertical.Theverticalline
through (4, 3) intersects the x-axisat(4,0).
You can immediately see that the distance from P to
line is3units,butyoucanalsousethedistance
formula to confirm.
Write a conjecture that describes the pattern in
each sequence. Then use your conjecture to
find the next item in the sequence.
61.
SOLUTION:
By comparing all these three items, the first item has
two triangles those are facing towards the right, the
second item has three triangles those are facing
upwards, the third item has four triangles those are
facing towards the right. By observing the items, the
next item should have five triangles; those should
face upwards.
62.
SOLUTION:
The first figure has 1 square block, the second figure
has 1 + 2 square blocks, and the third figure has 1 +
2 + 3 square blocks and arranges the blocks as
shown. So, the fourth figure has 1 + 2 + 3 + 4 square
blocks as below.
State the property that justifies each statement.
63.If thenx = 14.
SOLUTION:
Multiplication Property
64.If x = 5 and b = 5, then x = b.
SOLUTION:
Substitution Property
65.If XY – AB = WZ – AB, then XY = WZ.
SOLUTION:
Addition Property
66.If m A = m B and m B = m C, m A = m
C.
SOLUTION:
Transitive Property
67.If m1 + m2 = 90 and m2 = m3, then m1
+ m3 = 90.
SOLUTION:
Substitution Property
Find the measures of each numbered angle.
1.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. Let x be the measure of unknown angle in the
figure.
2.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
In the figure,
In the figure, andtheanglemeasuring39°are
congruent.
So,
Find each measure.
3.m2
SOLUTION:
BytheExteriorAngleTheorem, .
4.mMPQ
SOLUTION:
By the Exterior Angle Theorem,
DECK CHAIRS The brace of this deck chair
forms a triangle with the rest of the chair’s
frame as shown. If m1 = 102 and m3 = 53,
find each measure.
Refer to the figure on page 250.
5.m4
SOLUTION:
By the Exterior Angle Theorem,
Substitute.
6.m6
SOLUTION:
In the figure, and form a linear pair. So,
7.m2
SOLUTION:
By the Exterior Angle Theorem,
Substitute.
The sum of the measures of the angles of a triangle
is 180.
So,
Substitute.
8.m5
SOLUTION:
Angles 4 and 5 form a linear pair. Use the Exterior
Angle Theorem to find first and then use the
fact that the sum of the measures of the two angles
of a linear pair is 180.
By the Exterior Angle Theorem,
Substitute.
In the figure, and form a linear pair. So,
CCSSREGULARITYFind each measure.
9.m1
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
10.m3
SOLUTION:
In the figure, and angleformalinearpair.
So,
The sum of the measures of the angles of a triangle
is 180. So, .
Substitute.
11.m2
SOLUTION:
In the figure, and angleformalinearpair.
So,
Find the measure of each numbered angle.
12.Refer to the figure on page 250.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
13.Refer to the figure on page 250.
SOLUTION:
The sum of the measures of the angles of a triangle
is 180. So, .
14.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.In .
Here, and arecongruentangles.Bythe
definition of congruence,
In .
Since , .
15.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.In .
In the figure, and are vertical angles. Since
vertical angles are congruent, .
In .
Substitute.
16.AIRPLANES The path of an airplane can be
modeled using two sides of a triangle as shown. The
distance covered during the plane’s ascent is equal to
the distance covered during its descent.
a. Classify the model using its sides and angles.
b. The angles of ascent and descent are congruent.
Find their measures.
SOLUTION:
a. The triangle has two congruent sides. So, it is
isosceles. One angle of the triangle measures 173, so
it is a obtuse angle. Since the triangle has an obtuse
angle, it is an obtuse triangle.
b. Let x be the angle measure of ascent and descent.
We know that
the sum of the measures of the angles of a triangle is
180. So, .
The angle of ascent is 3.5 and the angle of descent
is 3.5.
Find each measure.
17.m1
SOLUTION:
By the Exterior Angle Theorem,
Find .
18.m3
SOLUTION:
By the Exterior Angle Theorem,
Find .
19.m2
SOLUTION:
By the Exterior Angle Theorem,
Solve for .
That is,
20.m4
SOLUTION:
By the Exterior Angle Theorem,
Solve for .
That is,
21.mABC
SOLUTION:
By the Exterior Angle Theorem,
Find .
That is, .
Substitute in
22.mJKL
SOLUTION:
By the Exterior Angle Theorem,
.
Find x.
That is, .
Substitute in
23.WHEELCHAIR RAMP Suppose the wheelchair
rampshownmakesa12°anglewiththeground.
What is the measure of the angle the ramp makes
with the van door?
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
Let x be the measure of the angle the ramp makes
with the van door.
CCSSREGULARITYFindeachmeasure.
24.m1
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
25.m2
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
26.m3
SOLUTION:
By the Exterior Angle Theorem, .
That is,
27.m4
SOLUTION:
In the figure,
The sum of the measures of the angles of a triangle
is180.
In the figure, .
Substitute.
28.m5
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
29.m6
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
In the figure,
Substitute.
ALGEBRA Find the value of x. Then find the
measure of each angle.
30.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
Solve for x.
Substitute ineachmeasure.
31.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
Solve for x.
Substitute in2x.
32.
SOLUTION:
By the Exterior Angle Theorem,
.
Solve for x.
Substitute in5x + 62.
Substitute in3x + 47.
33.GARDENING A landscaper is forming an isosceles
triangle in a flowerbed using chrysanthemums. She
wants m A to be three times the measure of B
and C. What should the measure of each angle
be?
SOLUTION:
and
The sum of the measures of the angles of a triangle
is 180. In the figure, .
Substitute.
Since , .
Substitute in .
PROOF Write the specified type of proof.
34.flow proof of Corollary 4.1
SOLUTION:
Given:
isarightangle.
Prove: and are complementary.
Proof:
35.paragraph proof of Corollary 4.2
SOLUTION:
Given:
is a right angle.
Prove: There can be at most one right angle in a
triangle.
Proof: In , isarightangle.
. so
.
If werearightangle,then . But that
is impossible, so there cannot be two right angles in a
triangle.
Given:
isobtuse.
Prove: There can be at most one obtuse angle in a
triangle.
Proof: In , isobtuse.So .
. It must be that
. So, and must be acute.
CCSSREGULARITYFind the measure of
each numbered angle.
36.
SOLUTION:
The sum of the measures of the angles of a triangle
is180.
In the figure, .
In the figure, .
Solve for
By the Exterior Angle Theorem, .
Substitute.
Also, .
Substitute.
37.
SOLUTION:
Look for pairs of vertical angles first. Here,
angle and are vertical angles, since
angle and are vertical angles, they are
congruent. By the definition of congruence,
.
Look for linear pairs next. Angles 5 and 7 are a
linear pair. Since ,
or70.
Next, the Triangle Angle Sum theorem can be used
to find .
From the diagram, . By the Exterior Angle
Theorem, .
Since congruent angles have equal measure,
.
Using the Triangle Angle Sum Theorem we know
that .
Using the Triangle Angle Sum Theorem we know
that .
Using the Triangle Angle Sum Theorem we know
that .
38.ALGEBRA Classify the triangle shown by its
angles. Explain your reasoning.
SOLUTION:
Obtuse; the sum of the measures of the three angles
of a triangle is 180. So, (15x + 1) + (6x + 5) + (4x –
1) = 180 and x = 7. Substituting 7 into the expressions
for each angle, the angle measures are 106, 47, and
27. Since the triangle has an obtuse angle, it is
obtuse.
39.ALGEBRA The measure of the larger acute angle
in a right triangle is two degrees less than three times
the measure of the smaller acute angle. Find the
measure of each angle.
SOLUTION:
Let x and y be the measure of the larger and smaller
acute angles in a right triangle respectively. Given
that . The sum of the measures of the
angles of a triangle is 180.
So,
Substitute.
Substitute in .
Thus the measure of the larger acute angle is 67 and
the measure of the smaller acute angle is 23.
40.Determine whether the following statement is true or
false. If false, give a counterexample. If true, give an
argument to support your conclusion.
If the sum of two acute angles of a triangle is
greater than 90, then the triangle is acute.
SOLUTION:
True; sample answer: Since the sum of the two acute
angles is greater than 90, the measure of the third
angle is a number greater than 90 subtracted from
180, which must be less than 90. Therefore, the
triangle has three acute angles and is acute.
41.ALGEBRA In , m X = 157, m Y = y, and
m Z = z. Write an inequality to describe the
possible measures of Z. Explain your reasoning.
SOLUTION:
z < 23; Sample answer: Since the sum of the
measures of the angles of a triangle is 180 and
, , so
If was0,then
wouldequal23.Butsinceananglemusthave
a measure greater than 0, mustbelessthan
23, so z < 23.
42.CARS Refer to the photo on page 252.
a. Find m1 and m2.
b. If the support for the hood were shorter than the
one shown, how would m1 change? Explain.
c. If the support for the hood were shorter than the
one shown, how would m2 change? Explain.
SOLUTION:
a. By the Exterior Angle Theorem,
So, Inthefigure,
Solve for .
b. Sample answer: The measure of wouldget
larger if the support were shorter because the hood
would be closer to the leg of the triangle that is along
the fender of the car.
c. Sample answer: The measure of wouldget
smaller if the support were shorter because
wouldgetlargerandtheyarealinearpair.
PROOF Write the specified type of proof.
43.two-column proof
Given: RSTUV is a pentagon.
Prove: m S + mSTU + mTUV + m V +
mVRS = 540
SOLUTION:
Proof: Statements (Reasons)
1. RSTUV is a pentagon. (Given)
2. ;
;
( Sum Thm.)
3. (Add. Prop.)
4. ;
;
( Addition)
5. (Subst.)
44.flow proof
Given: 3 5
Prove: m1 + m2 = m6 + m7
SOLUTION:
Proof:
45.MULTIPLE REPRESENTATIONS In this
problem, you will explore the sum of the measures of
the exterior angles of a triangle.
a. GEOMETRIC Draw five different triangles,
extending the sides and labeling the angles as shown.
Be sure to include at least one obtuse, one right, and
one acute triangle.
b. TABULAR Measure the exterior angles of each
triangle. Record the measures for each triangle and
the sum of these measures in a table.
c. VERBAL Make a conjecture about the sum of
the exterior angles of a triangle. State your
conjecture using words.
d. ALGEBRAIC State the conjecture you wrote in
part c algebraically.
e. ANALYTICAL Write a paragraph proof of your
conjecture.
SOLUTION:
a. Sample answer:
b. Sample answer:
c. Sample answer:
The sum of the measures of the exterior angles of a
triangle is 360.
d. m1 + m2 + m3 = 360
e. The Exterior Angle Theorem tells us that m3 =
mBAC + mBCA,
m2 = mBAC + mCBA,
m1 = mCBA + mBCA.
Through substitution,
m1+m2 + m3 = mCBA + mBCA +
mBAC + mCBA + mBAC + mBCA. Which
can be simplified to m1+m2 + m3 = 2m
BAC + 2mBCA + 2mCBA.
The Distributive Property can be applied and gives
m1+m2 + m3 = 2(mBAC + mBCA +
mCBA). The Triangle Angle-Sum Theorem tells us
that
mBAC + mBCA + mCBA = 180. Through
substitution we have m1+m2 + m3 = 2(180)
= 360.
46.CCSS CRITIQUE Curtis measured and labeled the
angles of the triangle as shown. Arnoldo says that at
least one of his measures is incorrect. Explain in at
least two different ways how Arnoldo knows that
this is true.
SOLUTION:
Sample answer: Corollary 4.2 states that there can
be at most one right or obtuse angle in a triangle.
Since this triangle is labeled with two obtuse angle
measures, 93 and 130, at least one of these measures
must be incorrect. Since by the Triangle Angle Sum
Theorem the sum of the interior angles of the triangle
mustbe180and37+93+130≠180,atleastoneof
these measures must be incorrect.
47.WRITING IN MATH Explain how you would find
the missing measures in the figure shown.
SOLUTION:
The measure of is the supplement of the exterior
angle with measure 110, so or 70.
Because the angles with measures b and c are
congruent, b = c . Using the Exterior Angle
Theorem, b + c = 110. By substitution, b + b = 110,
so 2b = 110 and b = 55. Because b = c, c = 55.
48.OPEN ENDED Construct a right triangle and
measure one of the acute angles. Find the measure
of the second acute angle using calculation and
explain your method. Confirm your result using a
protractor.
SOLUTION:
Sample answer:
I found the measure of the second angle by
subtractingthefirstanglefrom90°sincetheacute
angles of a right triangle are complementary.
49.CHALLENGE Find the values of y and z in the
figure.
SOLUTION:
In the figure, because they
are a linear pair and
because of the External Angle Theorem.
Simplify the equations and name them.
Subtract the equation (2) from (1).
Substitute in(1).
50.REASONING If an exterior angle adjacent to A
is acute, is acute,right,obtuse,orcanits
classification not be determined? Explain your
reasoning.
SOLUTION:
Obtuse; since the exterior angle is acute, the sum of
the remote interior angles must be acute, which
means the third angle must be obtuse. Therefore, the
triangle must be obtuse. Also, since the exterior angle
forms a linear pair with A, A must be obtuse since
two acute angles cannot be a linear pair.
51.WRITING IN MATH Explain why a triangle
cannot have an obtuse, acute, and a right exterior
angle.
SOLUTION:
Sample answer: Since an exterior angle is acute, the
adjacent angle must be obtuse. Since another exterior
angle is right, the adjacent angle must be right. A
triangle cannot contain both a right and an obtuse
angle because it would be more than 180 degrees.
Therefore, a triangle cannot have an obtuse, acute,
and a right exterior angle.
52.PROBABILITY Mr. Glover owns a video store
and wants to survey his customers to find what type
of movies he should buy. Which of the following
options would be the best way for Mr. Glover to get
accurate survey results?
A surveying customers who come in from 9 p.m.
until 10 p.m.
B surveying customers who come in on the weekend
C surveying the male customers
D surveying at different times of the week and day
SOLUTION:
The most accurate survey would ask a random
sampling of customers. Choices A, B, and C each
survey a specific group of customers. Choice D is a
random sample of customers so it will give Mr.
Glover the most accurate result.
53.SHORT RESPONSE Two angles of a triangle
havemeasuresof35°and80°.Describethepossible
values of the exterior angle measures of the triangle.
SOLUTION:
Sample answer: Since the sum of the measures of
the angles of a triangle is 180, the measure of the
third angle is 180 – (35 + 80) or 60. To find the
measures of the exterior angles, subtract each angle
measure from 180. The values for the exterior angle
ofthetriangleare100°,115°,and145°.
54.ALGEBRA Which equation is equivalent to 7x – 3
(2 – 5x) = 8x?
F 2x – 6 = 8
G 22x – 6 = 8x
H –8x – 6 = 8x
J22x + 6 = 8x
SOLUTION:
7x – 3(2 – 5x) = 8xOriginal equation
7x – 6 + 15x = 8xDistributive Property
22x – 6 = 8xSimplify.
So, the correct option is G.
55.SAT/ACT Joey has 4 more video games than
Solana and half as many as Melissa. If together they
have 24 video games, how many does Melissa have?
A 7
B 9
C 12
D 13
E 14
SOLUTION:
Let j, s, and m be the number of video games with
Joey, Solana, and Melissa respectively. Given
that , , and .
Substitute in .
Substitute in .
So, Melissa has 14 video games. The correct option
is E.
Classify each triangle as acute, equiangular,
obtuse, or right.
56.
SOLUTION:
Since all the angles are congruent, it is equiangular.
57.
SOLUTION:
One angle of the triangle measures 150, so it is an
obtuse angle. Since the triangle has an obtuse angle,
it is an obtuse triangle.
58.
SOLUTION:
One angle of the triangle measures 90, so it is a right
angle. Since the triangle has a right angle, it is a right
triangle.
COORDINATE GEOMETRY Find the
distance from P to .
59.Line containspoints(0,–2) and (1, 3). Point P has
coordinates (–4, 4).
SOLUTION:
Find the equation of the line Substitutethevalues
in the slope formula.
Then write the equation of this line using the point (1,
3).
Therefore, the equation of the line l is
Write an equation of the line w perpendicular to
through (–4, 4). Sincetheslopeofline is 5, the
slope of a line w is . Write the equation of line w
through (–4, 4) with slope 1.
Therefore, the equation of the line w is
Solve the system of equations to determine the point
of intersection. The left sides of the equations are the
same. So, equate the right sides and solve for x.
Use the value of x to find the value of y.
So,thepointofintersectionis(1,3)
Use the Distance Formula to find the distance
between the points (–4, 4) and (1, 3) .
Therefore, the distance between the two lines
is
60.Line contains points (–3, 0) and (3, 0). Point P has
coordinates (4, 3).
SOLUTION:
Here, line ishorizontal;infactitisthex-axis. So, a
line perpendicular to isvertical.Theverticalline
through (4, 3) intersects the x-axisat(4,0).
You can immediately see that the distance from P to
line is3units,butyoucanalsousethedistance
formula to confirm.
Write a conjecture that describes the pattern in
each sequence. Then use your conjecture to
find the next item in the sequence.
61.
SOLUTION:
By comparing all these three items, the first item has
two triangles those are facing towards the right, the
second item has three triangles those are facing
upwards, the third item has four triangles those are
facing towards the right. By observing the items, the
next item should have five triangles; those should
face upwards.
62.
SOLUTION:
The first figure has 1 square block, the second figure
has 1 + 2 square blocks, and the third figure has 1 +
2 + 3 square blocks and arranges the blocks as
shown. So, the fourth figure has 1 + 2 + 3 + 4 square
blocks as below.
State the property that justifies each statement.
63.If thenx = 14.
SOLUTION:
Multiplication Property
64.If x = 5 and b = 5, then x = b.
SOLUTION:
Substitution Property
65.If XY – AB = WZ – AB, then XY = WZ.
SOLUTION:
Addition Property
66.If m A = m B and m B = m C, m A = m
C.
SOLUTION:
Transitive Property
67.If m1 + m2 = 90 and m2 = m3, then m1
+ m3 = 90.
SOLUTION:
Substitution Property
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4-2 Angles of Triangles
