Question

it is given that quadrilateral abcd is a kite. we know that $overline{ad}congoverline{cd}$ by the definition of kite. by the kite diagonal theorem, $overline{ac}$ is congruent to $overline{bd}$. this means that angles aed and ced are right angles. we also see that $overline{ed}congoverline{ed}$ by the reflexive property. therefore, we have that $\triangle aedcong\triangle ced$ by ( ). aas hl sas

Explanation:

Step1: Identify given right - angled triangles

We have right - angled triangles $\triangle AED$ and $\triangle CED$ with right angles $\angle AED$ and $\angle CED$.

Step2: List known congruent parts

We know that $\overline{AD}\cong\overline{CD}$ (given by the definition of a kite) and $\overline{ED}\cong\overline{ED}$ (by the reflexive property). Here, $\overline{AD}$ and $\overline{CD}$ are the hypotenuses of right - angled triangles $\triangle AED$ and $\triangle CED$ respectively, and $\overline{ED}$ is a common leg.

Step3: Apply congruence theorem

The Hypotenuse - Leg (HL) congruence theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. Since $\overline{AD}\cong\overline{CD}$ (hypotenuses) and $\overline{ED}\cong\overline{ED}$ (leg), $\triangle AED\cong\triangle CED$ by HL.

Answer:

HL