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Transcript
Name
LESSON
4-2
Date
Class
Reteach
Angle Relationships in Triangles
According to the Triangle Sum Theorem, the sum of the angle
measures of a triangle is 180°.
*
—
mJ mK mL 62 73 45
180°
,
The corollary below follows directly from the Triangle Sum Theorem.
Corollary
—
—
+
Example
The acute angles of a right
triangle are complementary.
mC 90 39
51°
#
—
%
$
mC mE 90°
Use the figure for Exercises 1 and 2.
1. Find mABC.
!
47°
—
2. Find mCAD.
$
38°
—
—
"
#
Use RST for Exercises 3 and 4.
2
3. What is the value of x?
X—
14
X—
4. What is the measure of each angle?
mR 85°; mS 30°; mT 65°
4
X—
3
What is the measure of each angle?
7
!
-
—
,
—
.
5. L
"
5
#
6. C
49°
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
X—
6
7. W
39.8°
14
(90 x)°
Holt Geometry
Name
Date
Class
Reteach
LESSON
4-2
Angle Relationships in Triangles
An exterior angle of a triangle is formed by
one side of the triangle and the extension of
an adjacent side.
continued
remote
interior angles
exterior
angle
1 and 2 are the remote interior angles of
4 because they are not adjacent to 4.
Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of its remote interior angles.
m4 m1 m2
Third Angles Theorem
If two angles of one triangle are congruent
to two angles of another triangle, then
the third pair of angles are congruent.
Find each angle measure.
&
$
X—
—
—
*
#
'
(
8. mG
X —
—
"
!
9. mD
51°
41°
Find each angle measure.
,
.
5
0
X—
X—
1
X—
4
+
X —
-
10. mM and mQ
2
11. mT and mR
82°; 82°
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
3
33°; 33°
15
Holt Geometry
Name
Date
LESSON
4-2
Class
Name
Practice A
4-2
Use the figure for Exercises 1–3. Name all the angles that fit the definition of
each vocabulary word.
1. exterior angle
2. remote interior angles to �6
3. interior angle
�1, �4, �6
�2, �3
�2, �3, �5
�
�
�
For Exercises 4–7, fill in the blanks to complete each theorem or corollary.
equiangular
4. The measure of each angle of an
triangle is 60°.
180�
5. The sum of the angle measures of a triangle is
right
exterior angle
6. The acute angles of a
.
triangle are complementary.
7. The measure of an
of the measures of its remote interior angles.
2. The acute angles of right triangle ABC are congruent.
Find their measures.
of a triangle is equal to the sum
�
35°
�
�
�
20°
�
130°
60�
Find each angle measure.
40°
12. m�P
�
(5� � 1)°
�
35�
�
9. m�E and m�G
�
120�
�
44�; 44�
�
�
�
�
(9� � 9)°
�
11. In �ABC and �DEF, m�A � m�D and m�B � m�E. Find m�F if an exterior
angle at A measures 107�, m�B � (5x � 2)�, and m�C � (5x � 5)�.
���
���
Date
108�; 108�
(10� � 2)°
(6� � 4)°
�
����
33�; 66�; 81�
10. m�T and m�V
�
(5� � 4)°
12. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle
measures of the triangle.
110�
11
�
47�
7. m�PRS
�
14. When a person’s joint is injured, the person often goes through
rehabilitation under the supervision of a doctor or physical
therapist to make sure the joint heals well. Rehabilitation
involves stretching and exercises. The figure shows a leg
bending at the knee during a rehabilitation session. Use
what you know about triangles to find the angle measure that
the knee is bent from the horizontal (fully extended) position.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
60�
8. In �LMN, the measure of an exterior angle at N measures 99�.
m�L � _1_x � and m�M � _2_x�. Find m�L, m�M, and m�LNM.
3
3
�
13. m�VWY
Name
�
�
�
6. m�B
80°
�
23°
65�
35°
�
4-2
�
�
� 120°
�
11. m�L
�
�
Class
Holt Geometry
12
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
Practice C
LESSON
4-2
Angle Relationships in Triangles
Date
Reteach
Angle Relationships in Triangles
�
���
� 180°
�
2. Use this figure to write a flowchart proof that the
sum of the measures of the exterior angles of a
triangle, one at each vertex, is 360�.
Corollary
Reasons
1. Given
2. Construction
3. Triangle Sum Thm.
m�C � 90 � 39
� 51°
�
m�C � m�E � 90°
5. Angle Add. Post.
Use the figure for Exercises 1 and 2.
6. Subst.
1. Find m�ABC.
�
47°
2
3
�
�
���
�
4 1
���
���
Example
The acute angles of a right
triangle are complementary.
4. Add. Prop. of �
�����������������
�����������������
���������������
�����������
6
���
2. Find m�CAD.
5
�
38°
���
���
�
�
Use �RST for Exercises 3 and 4.
3. What is the value of x ?
�������������������
������������������
���������������
����������
������
3. Find the sum of the exterior angles, one at each vertex, of a quadrilateral.
m�R � 85°; m�S � 30°; m�T � 65°
�����
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
���
�
5. �L
�� �� ��
���������
�
�
�
�
interior � 540�; exterior � 360�
���
�
What is the measure of each angle?
360�
4. Use the techniques you developed in Exercises 1–3 to find the sums of the measures
of the interior angles and of the exterior angles, one at each vertex, of a pentagon.
13
����������
���������
4. What is the measure of each angle?
����������������������
5. A landscape artist plans to draw a pair of mountains.
He wants his drawing to be reasonably accurate, so
he takes some measurements and draws this figure.
Find x, y, and z.
�
14
�������������������
������������������
���������
����������������������
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
The corollary below follows directly from the Triangle Sum Theorem.
Statements
1. Quadrilateral ABCD
2. Draw AC.
3. m�D � m�DAC � m�DCA � 180°,
m�B � m�BAC � m�BCA � 180�
4. m�D � m�DAC � m�DCA �
m�B � m�BAC � m�BCA � 360�
5. m�DAC � m�BAC � m�DAB,
m�DCA � m�BCA � m�DCB
6. m�D � m�DAB � m�B �
m�DCB � 360�
Holt Geometry
Class
m�J � m�K � m�L � 62 � 73 � 45
�
55�
54�; 72�; 54�
According to the Triangle Sum Theorem, the sum of the angle
measures of a triangle is 180°.
1. Write a two-column proof that the sum of the angle measures of a quadrilateral
is 360�. Begin by drawing quadrilateral ABCD. (Hint: You will have to draw one
�
auxiliary line.)
�
Possible answer:
89.7�
5. 0.3�
(9� � 2)°
65°
�
�
LESSON
z�
4. (90 � z )�
�
10. m�G
�°
45.1�
3. 44.9�
70�
9. m�F
�
�
�
�
115�
8. m�B
45�
The measure of one of the acute angles in a right triangle is given. Find the
measure of the other acute angle.
Find the measure of each angle.
30°
Angle Relationships in Triangles
1. An area in central North Carolina is known as
the Research Triangle because of the relatively
�
Durham
large number of high-tech companies and research
10.7 mi
universities located there. Duke University, the
21.4 mi
Chapel
University of North Carolina at Chapel Hill, and
Hill
25.7 mi
North Carolina State University are all within this
Raleigh
area. The Research Triangle is roughly bounded
by the cities of Chapel Hill, Durham, and Raleigh.
From Chapel Hill, the angle between Durham and Raleigh
measures 54.8�. From Raleigh, the angle between Chapel Hill
and Durham measures 24.1�. Find the angle between
101.1�
Chapel Hill and Raleigh from Durham.
�
� �
Class
Practice B
LESSON
Angle Relationships in Triangles
�
Date
���
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
69
�
�
6. �C
49°
x � 93; y � 52; z � 35
�
��
�
7. �W
39.8°
14
(90 � x)°
Holt Geometry
Holt Geometry
Name
Date
Class
Name
Reteach
LESSON
4-2
Date
Challenge
LESSON
Angle Relationships in Triangles
An exterior angle of a triangle is formed by
one side of the triangle and the extension of
an adjacent side.
4-2
continued
remote
interior angles
Analyzing Verbal Descriptions and Using Auxiliary Figures
Find the measure of each angle.
exterior
angle
�1 and �2 are the remote interior angles of
�4 because they are not adjacent to �4.
�
88°
39°
_
m�4 � m�1 � m�2
�
�
_
BA � DE. Explain how you could draw one or more auxiliary figures
to help you find the value of x. Then find the value of x. Explain.
� �
3.
�
�
���
�
Third Angles Theorem
If two angles of one triangle are congruent
to two angles of another triangle, then
the third pair of angles are congruent.
�
�
�
�
_
�
���
����
�� �
�
�
�
�
41°
�
�
�
����������
����������
�
���������
�� �
�
�
11. m�T and m�R
33°; 33°
82°; 82°
15
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
LESSON
4-2
�
�
�
10. m�M and m�Q
Date
Class
Holt Geometry
Problem Solving
4. � lines � alt. int. � �
16
Holt Geometry
Class
Graphic Organizer
�°
155°
���������������
���������������������
����������������������������
����������������������������
�������������������������������
�����������������������������
�
�
�
�
��������������������
���������������
�
��������������������������������
�����������������������������
����������������������
�
�
Use the figure of the banner for Exercises 3 and 4.
���������
����������������������
3. What is the value of n?
�
n � 12
4. What is the measure of each angle in the banner?
�
����������������������
��������������������������
��������������������������������
���������������������������
���������������������������
���������
�
�
�
�
�
���
�
�
��
�
30°
�
�
60°
60°
30°
2. Find m�TRP.
3. Find m�RTS.
Use the figure for Exercises 4–7.
113°
�
The figure shows a path through a garden.
Choose the best answer.
�
1. Find m�QRP.
��
6. At takeoff, a ° � 23°. What is c °, the
measure of the angle the pole makes
with the athlete’s body?
32°
�
Use the given information to find the measures of the angles.
�S and �Q are right angles.
m�QPR � 30°
�TRP is equiangular.
��
�
�����������������������
�������������
���������������
Use the figure of the athlete pole vaulting for Exercises 5 and 6.
�������������������
���������������������������������
����������������������������������
���������������������������������������
���������������
�
���������
76°, 76°, 28°
�
80°
�
7. What is the measure of �QLP?
A 20°
C 110°
B 70°
D 125°
�
55°
30°
�
�
82°
55°
�
17
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
(2� � 20)°
(3� � 10)°
�
�
4. Find m�A.
�
5. Find m�B.
9. What is the measure of �PMN?
A 98°
C 60°
B 68°
D 55°
8. What is the measure of �LPM?
F 85°
H 95°
G 90°
J 125°
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Date
This graphic organizer describes the relationships of interior and exterior
angles in a triangle.
59°
5. What is x °, the measure of the angle that
the pole makes when it first touches
the ground?
�
�
Reading Strategies
4-2
146°
��
1
5. � Add. Post.
6. Subst. Prop. of �
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2. A large triangular piece of plywood is to be painted to look
like a mountain for the spring musical. The angles at the
base of the plywood measure 76° and 45°. What is the
measure of the top angle that represents the mountain peak?
��
�
5. m�BCD � m�3 � m�4
6. m�BCD � m�1 � m�2
LESSON
121°
�
2
4. m�2 � m�4
Name
Angle Relationships in Triangles
1. The locations of three food stands on a fair’s midway
are shown. What is the measure of the angle labeled x °?
�
�
Additional Answer: Proofs will vary.
Given: �ABC with exterior angle �BCD
4
3
1
Prove: m�BCD � m�1 � m�2
�
� �
Proof:
Statements
Reasons
1. �ABC with exterior angle �BCD 1. Given
2. Through
_ C, draw line � parallel 2. Through a point outside a line,
there is exactly one line parallel
to AB.
to the given line.
3. � lines � corr. � �
3. m�1 � m�3
���
�
Find each angle measure.
�
� ���
�
�
2
9. m�D
51°
�
4. It is possible to prove the Exterior Angle Theorem by drawing an
auxiliary line. Use the figure at the right to show how this might be done.
Write a complete proof on a separate sheet of paper. (Hint: Think
about the angles formed when parallel lines are cut by a transversal.)
�
���������
�
���
��� �
���
Through C, draw line � || BA. So � is also || DE. Then apply the Alt. Int.
� Thm. twice, followed by the � Add. Post. x° � 55° � 62° � 117°.
Find each angle measure.
�
�
����������� ���������
�������
�
���
�
�
8. m�G
2. In �WXY, the measure of �X is 4°
less than twice the measure of �W.
The measure of �Y is 24° less than
3 _1_ times m�W. Find m�Y.
2
1. In �FGH, the measure of �H is 14° less
than the measure of �F. The measure
of �G is 25° more than 2 _1_ times the
3
measure of �F. Find m�F.
�
Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of its remote interior angles.
Class
6. Find m�BCF.
7. Find m�EFD.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
70
�
40°
80°
120°
60°
18
Holt Geometry
Holt Geometry