See Problem 3.

13. Developing Proof Fill in the blanks to complete this proof of the Congruent Complements Theorem (Theorem 2-3).

    If two angles are complements of the same angle, then the two angles are congruent.

    

    Given: ∠ 1 angle 1  and ∠ 2 angle 2  are complementary.

    ∠ 3 angle 3  and ∠ 2 angle 2  are complementary.

    Prove: ∠ 1 ≅ ∠ 3 angle 1 approximately equal to , angle 3

    Proof: ∠ 1 angle 1  and ∠ 2 angle 2  are complementary and ∠ 3 angle 3  and ∠ 2 angle 2  are complementary because it is given. By the definition of complementary angles, m ∠ 1 + m ∠ 2 = m angle , 1 plus , m angle , 2 equals  a. ? ¯ modified question mark with under bar below  and m ∠ 3 + m ∠ 2 = m angle , 3 plus , m angle , 2 equals  b. ? ¯ . modified question mark with under bar below , .  Then m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 2 m angle , 1 plus , m angle , 2 equals , m angle , 3 plus , m angle 2  by the Transitive Property of Equality. Subtract m ∠ 2 m angle 2  from each side. By the Subtraction Property of Equality, you get m ∠ 1 = m angle , 1 equals  c. ? ¯ . modified question mark with under bar below , .  Angles with the same measure are d. ? ¯ , modified question mark with under bar below , comma  so ∠ 1 ≅ ∠ angle 1 approximately equal to angle  3.

B Apply

14. Think About a Plan What is the measure of the angle formed by Park St. and 116th St.?

    • Can you make a connection between the angle you need to find and the labeled angle?
    • How are angles that form a right angle related?

    

15. Open-Ended Give an example of vertical angles in your home or classroom.

Algebra Find the value of each variable and the measure of each labeled angle.

16. 
17. 

18. Developing Proof Fill in the blanks to complete this proof of Theorem 2-4. All right angles are congruent.

    

    Given: ∠ X angle x  and ∠ Y angle y  are right angles.

    Prove: ∠ X ≅ ∠ Y angle x , approximately equal to , angle y

    Proof: ∠ X angle x  and a. ? ¯ modified question mark with under bar below  are right angles because it is given.

    By the definition of b. ? ¯ , modified question mark with under bar below , comma   m ∠ X = 90 m angle , x equals , 90  and m ∠ Y = 90 . m angle , y equals , 90 .  By the Transitive Property of Equality, m ∠ X = m angle , x equals  c. ? ¯ . modified question mark with under bar below , .  Because angles of equal measure are congruent, d. ? ¯ . modified question mark with under bar below , .

19. Miniature Golf In the game of miniature golf, the ball bounces off the wall at the same angle it hit the wall. (This is the angle formed by the path of the ball and the line perpendicular to the wall at the point of contact.) In the diagram, the ball hits the wall at a 40 ° 40 degrees  angle. Using Theorem 2-3, what are the values of x and y?

    

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