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Kite Opposite Angles Theorem

Rule

Kite Opposite Angles Theorem

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

A kite with the congruent sides and angles marked

Based on the diagram above, the following relation holds true.

If is a kite and then and

Proof

Consider a kite such that and draw the diagonal This diagonal divides the kite into two triangles.

A kite with the longest diagonal drawn

By the Reflexive Property of Congruence, Then, the three sides of are congruent to the three sides of Therefore, by the Side-Side-Side Congruence Theorem. Thus, by definition of congruent polygons, To prove that use indirect reasoning and assume temporarily that these angles are congruent. Since then the opposite angles of the kite are congruent. Then, Parallelogram Opposite Angles Theorem tells that the quadrilateral is a parallelogram, which is a contradiction since it is a kite. This means that the temporary assumption is false implying that is not congruent to