Question

hk ⊥ jk, ∠jhk ≅ ∠hji, and ij ⊥ hi. complete the proof that ij ≅ hk.

1 hk ⊥ jk given
2 ∠jhk ≅ ∠hji given
3 ij ⊥ hi given
4 ∠i ≅ ∠k all right angles are congruent
5 hj ≅ hj reflexive property of congruence
6 △hjk ≅ △jhi
7 ij ≅ hk

Explanation:

Step1: Determine the congruence criterion

Since we have \(\angle JHK\cong\angle HJI\) (given), \(\angle I\cong\angle K\) (all right - angles are congruent), and \(\overline{HJ}\cong\overline{HJ}\) (reflexive property of congruence), by the AAS (Angle - Angle - Side) congruence criterion, \(\triangle HJK\cong\triangle JHI\).

Step2: Use the property of congruent triangles

If two triangles are congruent, then their corresponding sides are congruent. Since \(\triangle HJK\cong\triangle JHI\), the corresponding sides \(\overline{IJ}\) and \(\overline{HK}\) are congruent.

Answer:

1. Reason: AAS (Angle - Angle - Side) Congruence Criterion
2. Reason: Corresponding parts of congruent triangles are congruent (CPCTC)