We are asked to show that in a kite one pair of opposite angles are congruent and the other pair of opposite angles are not. Let's prove these two parts separately.
Congruent opposite angles
Let's draw a kite and mark the congruent sides. Let's also draw a diagonal and investigate the triangles we get.
Triangles △ JKM and △ LKM have three congruent sides, so according to the Side-Side-Side (SSS) Congruence Postulate they are congruent. The opposite angles of kite JKLM at J and L are corresponding angles in these triangles, so they are congruent.
∠ J≅∠ L
Not congruent opposite angles
Let's see what would follow if we assumed that angles ∠ K and ∠ M were also congruent. In this case, quadrilateral JKLM would have two pairs of congruent opposite angles, so according to Theorem 6.10 it would be a parallelogram.
According to Theorem 6.3, in a parallelogram opposite sides are congruent. However, by definition a kite does not have opposite sides that are congruent. This is a contradiction, so the assumption that angles ∠ K and ∠ M are congruent cannot be true.
∠ K≆∠ M
Let's summarize the two proofs above in a paragraph proof.
Completed Proof
2
&Given:&& JKLM is a kite
& && KJ≅KL
& && MJ≅ML
&Prove:&& ∠ J≅∠ L
& && ∠ K≆ ∠ M
Proof:
In triangles △ JKM and △ LKM, side KM is common and the other two corresponding sides are given as congruent. According to the SSS Postulate, these triangles are congruent, so corresponding angles ∠ J and ∠ L are congruent. If angles ∠ K and ∠ M were also congruent, then quadrilateral JKLM would be a parallelogram with congruent opposite sides. However, in kites opposite sides are not congruent, so JKLM is not a parallelogram. This means that ∠ K and ∠ M are not congruent.
