Download 4-2 Practice A Angle Relationships in Triangles

Name
LESSON
4-2
Date
Class
Practice A
Angle Relationships in Triangles
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
For Exercises 4–7, fill in the blanks to complete each theorem or corollary.
triangle is 60°.
4. The measure of each angle of an
.
5. The sum of the angle measures of a triangle is
triangle are complementary.
6. The acute angles of a
7. The measure of an
of the measures of its remote interior angles.
of a triangle is equal to the sum
Find the measure of each angle.
!
30°
#
35°
$
"
20°
&
8. m⬔B
9. m⬔F
'
(
,
130°
)
65°
*
10. m⬔G
+
11. m⬔L
0
X°
%
3
1
40°
35°
9
5
2
80°
4
6
12. m⬔P
7
8
13. m⬔VWY
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.
11
—
—
—
Holt Geometry
LESSON
4-2
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
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°
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.
For Exercises 4–7, fill in the blanks to complete each theorem or corollary.
!
Practice B
LESSON
Angle Relationships in Triangles
$
"
%
20°
45.1⬚
3. 44.9⬚
z⬚
4. (90 ⫺ z )⬚
89.7⬚
5. 0.3⬚
&
Find each angle measure.
115⬚
8. m⬔B
'
70⬚
9. m⬔F
(
130°
)
(9X 2)°
65°
*
60⬚
10. m⬔G
40°
35°
12. m⬔P
6
35⬚
1
47⬚
7. m⬔PRS
8. In 䉭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
9
7
44⬚; 44⬚
9. m⬔E and m⬔G
80°
4
60⬚
6. m⬔B
5
2
2
3
"
65⬚
3
1
+
11. m⬔L
0
23°
(5X 1)°
,
!
X°
0
$
# 120°
8
120⬚
4
5
&
7
(
(5X 4)°
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.
(9N 9)°
'
—
(10N 2)°
(6X 4)°
$
108⬚; 108⬚
10. m⬔T and m⬔V
%
13. m⬔VWY
33⬚; 66⬚; 81⬚
6
11. In 䉭ABC and 䉭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)⬚.
—
55⬚
—
12. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle
measures of the triangle.
54⬚; 72⬚; 54⬚
110⬚
11
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
LESSON
4-2
Holt Geometry
12
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Practice C
LESSON
4-2
Angle Relationships in Triangles
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.)
"
Holt Geometry
Review for Mastery
Angle Relationships in Triangles
According to the Triangle Sum Theorem, the sum of the angle
measures of a triangle is 180°.
J
62°
m⬔J ⫹ m⬔K ⫹ m⬔L ⫽ 62 ⫹ 73 ⫹ 45
⫽ 180°
Possible answer:
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⬚
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.
73°
45°
K
Example
The acute angles of a right
triangle are complementary.
m⬔C ⫽ 90 ⫺ 39
⫽ 51°
C
39°
E
4. Add. Prop. of ⫽
D
m⬔C ⫹ m⬔E ⫽ 90°
5. Angle Add. Post.
Use the figure for Exercises 1 and 2.
6. Subst.
1. Find m⬔ABC.
A
47°
49°
4 1
3
6
2. Find m⬔CAD.
2
D
5
MMM
MMM
MMM
%XT4HM
46°
84°
B
C
38°
Use 䉭RST for Exercises 3 and 4.
R
3. What is the value of x?
MMM
MMM
!DD0ROPOF
MMM
MMM
3IMPLIFY
(2x ⫹ 2)°
4. What is the measure of each angle?
MMM O3UM4HM
3UBST
3. Find the sum of the exterior angles, one at each vertex, of a quadrilateral.
M
13
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
S
W
A
50.2°
L
Y— X— Z—
(4x ⫹ 9)°
What is the measure of each angle?
360⬚
interior ⫽ 540⬚; exterior ⫽ 360⬚
—
T
m⬔R ⫽ 85°; m⬔S ⫽ 30°; m⬔T ⫽ 65°
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.
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.
(7x ⫺ 13)°
14
MMM Copyright © by Holt, Rinehart and Winston.
All rights reserved.
L
41°
N
5. ⬔L
—
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
69
001-082_Go08an_CRF_c04.indd 14
U
C
6. ⬔C
49°
x ⫽ 93; y ⫽ 52; z ⫽ 35
B
x°
V
7. ⬔W
39.8°
14
(90 ⫺ x)°
Holt Geometry
Holt Geometry
4/12/07 11:45:16 AM