Proof for Theorem
Prove: If a quadrilateral is a kite, it has one diagonal that bisects a pair of opposite angles.

thkite5given
thkite1
 
Statements
  
Reasons
1.
kite ABCD
1.
Given
2.
phkite3
thkite32
2.
A kite has 2 distinct sets of congruent adjacent sides.
3.
bdcongruent bd
3.
Reflexive property.
4.
thkite44
4.
SSS - if 3 sides of 1 Δ are congruent to the corres. parts of other Δ, the Δs are congruent.
5.
∠ABD congruent ∠CBD
∠ADB congruent ∠CDB
5.
CPCTC - Corres. parts of congruent triangles are congruent.
6.
thkite56
6.
An ∠ bisector is a ray in the interior of an ∠ forming two congruent angles.

There is only ONE diagonal that bisects the opposite angles.
If diagonal ac is drawn, ∠BAC is not congruent to ∠DAC (and ∠BCA is not congruent to ∠DCA). Since thkite3addby the definition of a kite, ΔABC is not congruent to ΔADC.

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