Download 2-8

NAME ______________________________________________ DATE
2-8
____________ PERIOD _____
Study Guide and Intervention
Proving Angle Relationships
Supplementary and Complementary Angles There are two basic postulates for
working with angles. The Protractor Postulate assigns numbers to angle measures, and the
Angle Addition Postulate relates parts of an angle to the whole angle.
Protractor
Postulate
and a number r between 0 and 180, there is exactly one ray
Given AB
, such that the measure
with endpoint A, extending on either side of AB
of the angle formed is r.
Angle Addition
Postulate
R is in the interior of PQS if and only if
mPQR mRQS mPQS.
P
R
Q
S
The two postulates can be used to prove the following two theorems.
If two angles form a linear pair, then they are supplementary angles.
If 1 and 2 form a linear pair, then m1 m2 180.
D
Complement
Theorem
If the noncommon sides of two adjacent angles form a right angle,
then the angles are complementary angles.
⊥ GH
, then m3 m4 90.
If GF
Example 1
B
C
F
J
3
4
G
H
Example 2
If 1 and 2 form a
linear pair and m2 115, find m1.
If 1 and 2 form a
right angle and m2 20, find m1.
R
Q
W
2
2 1
M
2
1
A
N
1
S
P
m1 m2 180
m1 115 180
m1 65
T
m1 m2 90
m1 20 90
m1 70
Suppl. Theorem
Substitution
Subtraction Prop.
Compl. Theorem
Substitution
Subtraction Prop.
Exercises
Find the measure of each numbered angle.
1.
2.
T
P
7
Q
8
8
R
S
m7 5x 5,
m8 x 5
©
Y
X
Glencoe/McGraw-Hill
U
7 6
Z
5
V
W
m5 5x, m6 4x 6,
m7 10x,
m8 12x 12
99
3. A
13
F
11
12
H
J
C
m11 11x,
m12 10x 10
Glencoe Geometry
Lesson 2-8
Supplement
Theorem
NAME ______________________________________________ DATE
2-8
____________ PERIOD _____
Study Guide and Intervention
(continued)
Proving Angle Relationships
Congruent and Right Angles
Three properties of angles can be proved as theorems.
Congruence of angles is reflexive, symmetric, and transitive.
Angles supplementary to the same angle
or to congruent angles are congruent.
Angles complementary to the same angle or to
congruent angles are congruent.
C
K
1
F
B
A
2
D
E
H
G
T
N
4
3
M
J
L
Q
If 1 and 2 are supplementary to 3,
then 1 2.
R
6
5
P
S
If 4 and 5 are complementary to 6,
then 4 5.
Example
Write a two-column proof.
Given: ABC and CBD are complementary.
DBE and CBD form a right angle.
Prove: ABC DBE
Statements
1. ABC and CBD are complementary.
DBE and CBD form a right angle.
2. DBE and CBD are complementary.
3. ABC DBE
C
A
D
B
E
Reasons
1. Given
2. Complement Theorem
3. Angles complementary to the same are .
Exercises
Complete each proof.
1. Given: A
B
⊥B
C
;
1 and 3 are
complementary.
Prove: 2 3
©
A
1 2
B
3
2. Given: 1 and 2
form a linear pair.
m1 m3 180
Prove: 2 3
E
C
D
T
L
1 2
B 3
R
P
Statements
Reasons
Statements
Reasons
a. A
B
⊥B
C
b.
a.
b. Definition of ⊥
a. 1 and 2 form
a linear pair.
m1 m3 180
a. Given
c. m1 m2 mABC
d. 1 and 2 form
a rt .
e. 1 and 2 are
compl.
f.
c.
b.
b. Suppl.
Theorem
d.
c. 1 is suppl.
to 3.
c.
e.
d.
d. suppl. to the
same are .
f. Given
g. 2 3
g.
Glencoe/McGraw-Hill
100
Glencoe Geometry
NAME ______________________________________________ DATE
2-8
____________ PERIOD _____
Skills Practice
Proving Angle Relationships
Find the measure of each numbered angle.
2. m5 22
1 2
4. m13 4x 11,
m14 3x 1
3. m1 38
5
2
1
6
5. 9 and 10 are
complementary.
7 9, m8 41
6. m2 4x 26,
m3 3x 4
2
13 14
7
8 9
10
3
Determine whether the following statements are always, sometimes, or never true.
7. Two angles that are supplementary form a linear pair.
8. Two angles that are vertical are adjacent.
9. Copy and complete the following proof.
Given: QPS TPR
Prove: QPR TPS
Proof:
R
Q
Statements
Reasons
a.
a.
b. mQPS mTPR
b.
c. mQPS mQPR mRPS
c.
S
T
P
mTPR mTPS mRPS
©
d.
d. Substitution
e.
e.
f.
f.
Glencoe/McGraw-Hill
101
Glencoe Geometry
Lesson 2-8
1. m2 57
NAME ______________________________________________ DATE
2-8
____________ PERIOD _____
Practice
Proving Angle Relationships
Find the measure of each numbered angle.
1. m1 x 10
m2 3x 18
2. m4 2x 5
m5 4x 13
3. m6 7x 24
m7 5x 14
4
3 5
1
6
7
2
Determine whether the following statements are always, sometimes, or never true.
4. Two angles that are supplementary are complementary.
5. Complementary angles are congruent.
6. Write a two-column proof.
Given: 1 and 2 form a linear pair.
2 and 3 are supplementary.
Prove: 1 3
1 2
3
7. STREETS Refer to the figure. Barton Road and Olive Tree Lane
form a right angle at their intersection. Tryon Street forms a 57°
angle with Olive Tree Lane. What is the measure of the acute angle
Tryon Street forms with Barton Road?
©
Glencoe/McGraw-Hill
102
Barton
Rd
Tryon
St
Olive Tree Lane
Glencoe Geometry
NAME ______________________________________________ DATE
2-8
____________ PERIOD _____
Reading to Learn Mathematics
Proving Angle Relationships
Pre-Activity
How do scissors illustrate supplementary angles?
Read the introduction to Lesson 2-8 at the top of page 107 in your textbook.
Is it possible to open a pair of scissors so that the angles formed by the two
blades, a blade and a handle, and the two handles, are all congruent? If so,
explain how this could happen.
Reading the Lesson
1. Complete each sentence to form a statement that is always true.
a. If two angles form a linear pair, then they are adjacent and
.
.
c. If D is a point in the interior of ABC, then mABC mABD and a number x between
d. Given RS
and
, there is exactly one ray
with endpoint R, extended on either side of RS, such that the measure of the angle
formed is x.
.
Lesson 2-8
b. If two angles are complementary to the same angle, then they are
e. If two angles are congruent and supplementary, then each angle is a(n)
angle.
f.
lines form congruent adjacent angles.
g. “Every angle is congruent to itself” is a statement of the
of angle congruence.
Property
h. If two congruent angles form a linear pair, then the measure of each angle is
.
i. If the noncommon sides of two adjacent angles form a right angle, then the angles are
.
2. Determine whether each statement is always, sometimes, or never true.
a. Supplementary angles are congruent.
b. If two angles form a linear pair, they are complementary.
c. Two vertical angles are supplementary.
d. Two adjacent angles form a linear pair.
e. Two vertical angles form a linear pair.
f. Complementary angles are congruent.
g. Two angles that are congruent to the same angle are congruent to each other.
h. Complementary angles are adjacent angles.
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose that a
classmate thinks that two angles can only be vertical angles if one angle lies above the
other. How can you explain to him the meaning of vertical angles, using the word vertex
in your explanation?
©
Glencoe/McGraw-Hill
103
Glencoe Geometry