Name two pairs of congruent angles in each figure. Justify your answers.

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23. Developing Proof Fill in the blanks to complete this proof of Theorem 2-5.

    If two angles are congruent and supplementary, then each is a right angle.

    Given: ∠ W angle w  and ∠ V angle v  are congruent and supplementary.

    Prove: ∠ W angle w  and ∠ V angle v  are right angles.

    

    Proof: ∠ W angle w  and ∠ V angle v  are congruent because a. ? ¯ . modified question mark with under bar below , .  Because congruent angles have the same measure, m ∠ W = m angle , w equals  b. ? ¯ . modified question mark with under bar below , .   ∠ W angle w  and ∠ V angle v  are supplementary because it is given. By the definition of supplementary angles, m ∠ W + m ∠ V = m angle , w plus , m angle , v equals  c. ? ¯ . modified question mark with under bar below , .  Substituting m ∠ W m angle w  for m ∠ V , m angle v comma  you get m ∠ W + m ∠ W = 180 , m angle , w plus , m angle , w equals , 180 comma  or 2 m ∠ W = 180 . 2 m angle , w equals , 180 .  By the d. ? ¯ modified question mark with under bar below  Property of Equality, m ∠ W = 90 . m angle , w equals , 90 .  Since m ∠ W = m ∠ V , m angle , w equals , m angle v comma   m ∠ V = 90 m angle , v equals , 90  by the Transitive Property of Equality. Both angles are e. ? ¯ modified question mark with under bar below  angles by the definition of right angles.

24. Design In the photograph, the legs of the table are constructed so that ∠ 1 ≅ ∠ 2 . angle 1 approximately equal to , angle 2 , .  What theorem can you use to justify the statement that ∠ 3 ≅ ∠ 4 ? angle 3 approximately equal to , angle 4 , question mark

    

25. Reasoning Explain why this statement is true: If m ∠ A B C + m ∠ X Y Z = 180 m angle eh b c plus m angle x y z equals 180  and ∠ A B C ≅ ∠ X Y Z , angle eh b c approximately equal to angle x y z comma  then ∠ A B C angle eh b c  and ∠ X Y Z angle x y z  are right angles.

Algebra Find the measure of each angle.

26. ∠ A angle eh  is twice as large as its complement, ∠ B . angle b , .
27. ∠ A angle eh  is half as large as its complement, ∠ B . angle b , .
28. ∠ A angle eh  is twice as large as its supplement, ∠ B . angle b , .
29. ∠ A angle eh  is half as large as twice its supplement, ∠ B . angle b , .

30. Proof Write a proof for this form of Theorem 2-2.

    If two angles are supplements of congruent angles, then the two angles are congruent.

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

    ∠ 3 angle 3  and ∠ 4 angle 4  are supplementary.

    ∠ 2 ≅ ∠ 4 angle 2 approximately equal to , angle 4

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

    

C Challenge

31. Coordinate Geometry ∠ D O E angle d o e  contains points D(2, 3), O(0, 0), and E(5, 1). Find the coordinates of a point F so that O F → o f vector  is a side of an angle that is adjacent and supplementary to ∠ D O E . angle d o e .

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