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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: 3
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EXAMPLE 1 Use right angle congruence
GIVEN: ABBC , DC BC
PROVE: B C
Write a proof.
STATEMENT REASONS
1.Given
2.Definition of perpendicularlines
3.Right Angles CongruenceTheorem
2. B and C are right angles.
3. B C
1.ABBC , DC BC
EXAMPLE 2 Prove a case of Congruent Supplements Theorem
GIVEN: 1 and 2 are supplements.3 and 2 are supplements.
PROVE: 1 3
Prove that two angles supplementary to the same angle are congruent.
EXAMPLE 2 Prove a case of Congruent Supplements Theorem
STATEMENT REASONS
1.3 and 2 are supplements.1 and 2 are supplements. Given1.
2. m 1+ m 2 = 180°m 3+ m 2 = 180°
2. Definition of supplementary angles
Transitive Property of Equality
3.3. m 1 + m 2 = m 3 + m 2
4. m 1 = m 3
5. 1 3
Subtraction Property of Equality
4.
Definition of congruent angles
5.
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?
2. Draw a diagram and write GIVEN and PROVE statements for a proof of each case of the Congruent Complements Theorem.
ANSWER2 Steps
GUIDED PRACTICE for Examples 1 and 2
Write a proof.
Given: 1 and 3 are complements; 3 and 5 are complements.
Prove: ∠1 5
ANSWER
GUIDED PRACTICE for Examples 1 and 2
Statements (Reasons)
1. 1 and 3 are complements; 3 and 5 are complements.
(Given)
2. ∠1 5Congruent Complements Theorem.
EXAMPLE 3Prove the Vertical Angles Congruence Theorem
GIVEN: 5 and 7 are vertical angles.
PROVE:∠5 ∠7
Prove vertical angles are congruent.
EXAMPLE 3Prove the Vertical Angles Congruence Theorem
5 and 7 are vertical angles.1.
STATEMENT REASONS
1.Given
2. 5 and 7 are a linear pair. 6 and 7 are a linear pair.
2.Definition of linear pair, as shown in the diagram
3. 5 and 7 are supplementary. 6 and 7 are supplementary.
3.Linear Pair Postulate
4.∠5 ∠7 Congruent Supplements Theorem
4.
GUIDED PRACTICE for Example 3
In Exercises 3–5, use the diagram.
3. If m 1 = 112°, find m 2, m 3, and m 4.
ANSWERm 2 = 68°
m 3 = 112° m 4 = 68°
GUIDED PRACTICE for Example 3
4. If m 2 = 67°, find m 1, m 3, and m 4.
ANSWERm 1 = 113°
m 3 = 113° m 4 = 67°
5. If m 4 = 71°, find m 1, m 2, and m 3.
ANSWERm 1 = 109°
m 2 = 71° m 3 = 109°
GUIDED PRACTICE for Example 3
6. Which previously proven theorem is used in Example 3 as a reason?
Congruent Supplements TheoremANSWER
EXAMPLE 4 Standardized Test Practice
SOLUTIONBecause TPQ and QPR form a linear pair, the sum of their measures is 180.
The correct answer is B.
ANSWER
GUIDED PRACTICE for 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 Original equation
32 + 3x +1 = 180 Distributive property of equality
3x = 147 Subtract 33 from each side
x = 49 Divide each side by 3
Use the diagram in Example 4.
GUIDED PRACTICE for Example 4
8. Find m TPS.
m TPS = (3x + 1)°
Substitute the value x = 49
m TPS = (147 +1)°
m TPS = 148°
SOLUTION
Use the diagram in Example 4.
m TPS = (3 49 +1)°