29. Reasoning Can a rhombus that is not a square be inscribed in a circle? Justify your answer.
30. Constructions The diagrams below show the construction of a tangent to a circle from a point outside the circle. Explain why B C ↔ modified b c with left right arrow above  must be tangent to ⊙ A . circle dot eh .  (Hint: Copy the third diagram and draw A C ¯ . eh c bar , . )

    Given: ⊙ A circle dot eh  and point B Construct the midpoint of A B ¯ . eh b bar , .  Label the point O.

    Construct a semicircle with radius OA and center O. Label its intersection with ⊙ A circle dot eh  as C.

    Draw B C ↔ . modified b c with left right arrow above , .

Write a proof for Exercises 31–34.

31. Proof Inscribed Angle Theorem, Corollary 1

    Given: ⊙ O , ∠ A circle dot o comma angle eh  intercepts B C ⌢ , ∠ D modified b c with frown above , comma . angle d  intercepts B C ⌢ . modified b c with frown above , .

    Prove: ∠ A ≅ ∠ D angle , eh approximately equal to , angle d

    

32. Proof Inscribed Angle Theorem, Corollary 2

    Given: ⊙ O circle dot o  with ∠ C A B angle c eh b  inscribed in a semicircle

    Prove: ∠ C A B angle c eh b  is a right angle.

    
33. Proof Inscribed Angle Theorem, Corollary 3

    Given: Quadrilateral ABCD inscribed in ⊙ O circle dot o

    Prove: ∠ A angle eh  and ∠ C angle c  are supplementary. ∠ B angle b  and ∠ D angle d  are supplementary.

    
34. Proof Theorem 12-12

    Given: G H ¯ g h bar  and tangent l intersecting ⊙ E circle dot e  at H

    Prove: m ∠ G H I = 1 2 m G F H ⌢ m angle g h i equals , 1 half , m . modified g f h with frown above

    

C Challenge

Reasoning Is the statement true or false? If it is true, give a convincing argument. If it is false, give a counterexample.

35. If two angles inscribed in a circle are congruent, then they intercept the same arc.
36. If an inscribed angle is a right angle, then it is inscribed in a semicircle.
37. A circle can always be circumscribed about a quadrilateral whose opposite angles are supplementary.

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