Question

k is the midpoint of (overline{ij}) and (overline{hk} perp overline{ij}). complete the proof that (angle j cong angle i).

(image of triangle hij with k on ij, hk drawn)

statement	reason
2. (overline{hk} perp overline{ij})	given
3. (angle hki cong angle hkj)	all right angles are congruent
4. (overline{ik} cong overline{jk})	definition of midpoint
5. (overline{hk} cong overline{hk})	reflexive property of congruence
6. (\triangle hik cong \triangle hjk)

Explanation:

Step1: Identify congruent parts

We have $\angle HKI \cong \angle HKJ$, $\overline{IK} \cong \overline{JK}$, and $\overline{HK} \cong \overline{HK}$.

Step2: Match congruence criterion

These are two sides and the included angle of $\triangle HIK$ and $\triangle HJK$ that are congruent, which fits the SAS (Side-Angle-Side) Congruence Postulate.

Step3: Deduce angle congruence

Since $\triangle HIK \cong \triangle HJK$, corresponding angles $\angle J$ and $\angle I$ are congruent (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Answer:

For statement 6, the reason is SAS (Side-Angle-Side) Congruence Postulate
Final conclusion: $\boldsymbol{\angle J \cong \angle I}$ (by CPCTC, after proving $\triangle HIK \cong \triangle HJK$)