Download - Use right angle congruence

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


Write a proof.

Given: 1 and 3 are complements; 3 and 5 are complements.

Prove: ∠1 5

ANSWER


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°


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°


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


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.


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)°