Exercise 15 Page 525

Let's take a look at the diagram we have been given.

Examining the diagram we see two pairs of vertical angles and four linear pairs.

cc Vertical Angles & Linear Pair ∠ 1 and ∠ 4 & ∠ 1 and ∠ 2 ∠ 2 and ∠ 3 & ∠ 1 and ∠ 3 & ∠ 2 and ∠ 4 & ∠ 3 and ∠ 4

By the Vertical Angles Theorem we know that the vertical angles are congruent. Therefore, it must be that ∠ 1 ≅ ∠ 4 and ∠ 2 ≅ ∠ 3 . Let's add this information to the diagram.

By the Linear Pair Postulate we know that linear pairs are supplementary. m∠ 1+ m∠ 2=180^(∘) m∠ 1+ m∠ 3=180^(∘) Using the Congruent Supplements Theorem we know that ∠ 2 and ∠ 3 are supplementary as well. Since ∠ 2 ≅ ∠ 3, they must both be right angles.

Statement	Reason
1. a ⊥ b	1. Given
2. ∠ 1 is a right angle	2. Definition of perpendicular lines
3. ∠ 1≅ ∠ 4	3. Vertical Angles Theorem
4. m∠ 1=90^(∘)	4. Definition of a right angle
5. m∠ 4=90^(∘)	5. Transitive Property of Equality
6. ∠ 1 and ∠ 2 are a linear pair	6. Definition of linear pair
7. ∠ 1 and ∠ 2 are supplementary	7. Linear Pair Postulate
8. m∠ 1+m∠ 2=180^(∘)	8. Definition of supplementary angles
9. 90^(∘)+m∠ 2=180^(∘)	9. Substitution Property of Equality
10. m∠ 2=90^(∘)	10. Subtraction Property of Equality
11. ∠ 2=∠ 3	11. Vertical Angles Theorem
12. m∠ 3=90^(∘)	12. Transitive Property of Equality
13. ∠ 1, ∠ 2, ∠ 3, ∠ 4 are right angles	13. Definition of a right angle

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