Question

hk bisects ∠ihj. complete the proof that ∠hkj ≅ ∠hki.

statement
1 hk bisects ∠ihj
2 hj ≅ hi
3 ∠ihk ≅ ∠jhk
4 hk ≅ hk
5

reason
given
given
definition of angle bisector
reflexive property of congruence

Explanation:

Step1: Use SAS Congruence Criterion

Since \(\overrightarrow{HK}\) bisects \(\angle IHJ\) (\(\angle IHK\cong\angle JHK\), statement 3), \(\overline{HJ}\cong\overline{HI}\) (statement 2), and \(\overline{HK}\cong\overline{HK}\) (statement 4), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle HKI\cong\triangle HKJ\).

Step2: Use Corresponding Parts of Congruent Triangles

If \(\triangle HKI\cong\triangle HKJ\), then their corresponding parts are congruent. So, \(\angle HKJ\cong\angle HKI\) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Answer:

1. \(\triangle HKI\cong\triangle HKJ\) ; SAS Congruence Criterion