EXAMPLE 1
Use right angle congruence
Write a proof.
AB
GIVEN:
B
PROVE:
BC , DC
C
REASONS
STATEMENT
1. AB
2.
3.
BC , DC
B and
angles.
B
BC
BC
C are right
1. Given
2. Definition of perpendicular
lines
C
3.Right Angles Congruence
Theorem
EXAMPLE 2
Prove a case of Congruent Supplements Theorem
Prove that two angles supplementary to the same angle
are congruent.
GIVEN: 1 and 2 are supplements.
3 and 2 are supplements.
PROVE:
1
3
EXAMPLE 2
Prove a case of Congruent Supplements Theorem
REASONS
STATEMENT
1.
1 and
3 and
2 are supplements. 1. Given
2 are supplements.
2. m 1+ m 2 = 180°
m 3+ m 2 = 180°
1+m
3. m
2= m
2. Definition of
supplementary angles
3+m
2 3. Transitive Property of
Equality
4. m
1=m
3
4. Subtraction
Property of Equality
5.
1
3
5. Definition of
congruent angles
GUIDED PRACTICE
for Examples 1 and 2
1. How many steps do you save in the proof in
Example 1 by using the Right Angles Congruence
Theorem?
ANSWER
2 Steps
2.
Draw a diagram and write GIVEN and PROVE
statements for a proof of each case of the
Congruent Complements Theorem.
GUIDED PRACTICE
for Examples 1 and 2
ANSWER
Write a proof.
Given:
1 and
3 and
Prove:  1
3 are complements;
5 are complements.
5
GUIDED PRACTICE
for Examples 1 and 2
Statements (Reasons)
1.
1 and
3 and
3 are complements;
5 are complements.
(Given)
2.  1
5 Congruent Complements Theorem.
EXAMPLE 3
Prove the Vertical Angles Congruence Theore
Prove vertical angles are congruent.
GIVEN:
5 and
7 are vertical angles.
PROVE:  5  7
EXAMPLE 3
Prove the Vertical Angles Congruence Theore
STATEMENT
REASONS
1.
5 and
7 are vertical angles. 1.Given
2.
5 and
6 and
7 are a linear pair.
7 are a linear pair.
5 and
6 and
7 are supplementary. 3.Linear Pair Postulate
7 are supplementary.
3.
4.  5  7
2.Definition of linear pair,
as shown in the
diagram
4.Congruent
Supplements Theorem
GUIDED PRACTICE
for Example 3
In Exercises 3–5, use the diagram.
3.
If m
1 = 112°, find m
ANSWER
m
2 = 68°
m
3 = 112°
m
4 = 68°
2, m
3, and m
4.
GUIDED PRACTICE
4.
If m
for Example 3
2 = 67°, find m
1, m
3, and m
4.
1, m
2, and m
3.
ANSWER
m
1 = 113°
5.
m
3 = 113°
m
4 = 67°
If m
4 = 71°, find m
ANSWER
m
1 = 109°
m
2 = 71°
m
3 = 109°
GUIDED PRACTICE
6.
for Example 3
Which previously proven theorem is used in
Example 3 as a reason?
ANSWER
Congruent Supplements Theorem
EXAMPLE 4
Standardized Test Practice
SOLUTION
Because TPQ and QPR form a linear pair, the sum
of their measures is 180.
ANSWER
The correct answer is B.
GUIDED PRACTICE
for Example 4
Use the diagram in Example 4.
7.
Solve for x.
SOLUTION
Because TPQ and QPR form a linear pair, the sum
of their measures is 180°. The correct answer is B.
32 + (3x +1) = 180
32 + 3x +1 = 180
3x = 147
x = 49
Original equation
Distributive property of equality
Subtract 33 from each side
Divide each side by 3
GUIDED PRACTICE
for Example 4
Use the diagram in Example 4.
8. Find m
TPS.
SOLUTION
m
TPS = (3x + 1)°
Substitute the value x = 49
m
TPS = (3 49 +1)°
m
TPS = (147 +1)°
m
TPS = 148°