Kite Opposite Angles Theorem

Consider a kite $A B C D$ such that $\overline{B A}$ is congruent to $\overline{B C}$ and draw the diagonal $\overline{B D} .$ This diagonal divides the kite into two triangles.

By the Reflexive Property of Congruence,

\overline{B D}

is congruent to itself. Then, the three sides of

△ A B D

are congruent to the three sides of

△ C B D .

\overline{B D} ≅ \overline{B D} \overline{B A} ≅ \overline{B C} \overline{A D} ≅ \overline{C D}

Therefore,

△ A B D

is congruent to

△ C B D

by the Side-Side-Side Congruence Theorem. Thus, by definition of congruent polygons,

∠ A

is congruent to

∠ C .

△ A B C ∠ A ≅ △ C B D ⇓ ≅ ∠ C

To prove that

∠ B

and

∠ D

are not congruent, use indirect reasoning and assume temporarily that these angles are congruent. That is, suppose that

∠ B

and

∠ D

are congruent. From the previous part,

∠ A

and

∠ C

are also congruent.

By the Converse Parallelogram Opposite Angles Theorem, $A B C D$ is a parallelogram. This contradicts the definition of a kite, which means that the temporary assumption is false. Therefore, $∠ B$ is not congruent to $∠ D .$ In conclusion, a kite has exactly one pair of congruent, opposite angles.

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Kite Opposite Angles Theorem

Proof

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