Question

$\angle jhk \cong \angle hji$, $\overline{ij} \perp \overline{hi}$, and $\overline{hk} \perp \overline{jk}$. complete the proof that $\overline{ij} \cong \overline{hk}$.

	statement	reason
2	$\overline{ij} \perp \overline{hi}$	given
3	$\overline{hk} \perp \overline{jk}$	given
4	$\angle i \cong \angle k$	all right angles are congruent
5	$\overline{hj} \cong \overline{hj}$	reflexive property of congruence
6	$\triangle hjk \cong \triangle jhi$
7	$\overline{ij} \cong \overline{hk}$	cpctc

Explanation:

Step1: Identify triangle congruence criteria

We have $\angle JHK \cong \angle HJI$ (given), $\angle I \cong \angle K$ (right angles congruent), and $\overline{HJ} \cong \overline{HJ}$ (reflexive property). This matches the AAS (Angle-Angle-Side) congruence rule, which states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, the triangles are congruent.

Step2: Apply congruence to the triangles

For $\triangle HJK$ and $\triangle JHI$:

• $\angle JHK \cong \angle HJI$ (Given)
• $\angle K \cong \angle I$ (Right angles congruent)
• $\overline{HJ} \cong \overline{HJ}$ (Reflexive Property)

By AAS, $\triangle HJK \cong \triangle JHI$.

Answer:

Angle-Angle-Side (AAS) Congruence Theorem