Question

∠hji ≅ ∠jhk and ∠hjk ≅ ∠ihj. complete the proof that △hij ≅ △jkh.
1 ∠hji ≅ ∠jhk given
2 ∠hjk ≅ ∠ihj given
3 \overline{hj} ≅ \overline{hj}
4 △hij ≅ △jkh

Explanation:

Step1: Identify the property of side \( \overline{HJ} \)

A side is congruent to itself. This is the reflexive property of congruence. So, \( \overline{HJ}\cong\overline{HJ} \) because of the reflexive property of congruence.

Step2: Apply the ASA (Angle - Side - Angle) congruence criterion

The ASA congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
In \( \triangle HIJ \) and \( \triangle JKH \):

• \( \angle HJI\cong\angle JHK \) (given)
• \( \overline{HJ}\cong\overline{HJ} \) (reflexive property)
• \( \angle HJK\cong\angle IHJ \) (given)

So, \( \triangle HIJ\cong\triangle JKH \) by the ASA (Angle - Side - Angle) congruence criterion.

Answer:

1. \( \angle HJI\cong\angle JHK \): Given
2. \( \angle HJK\cong\angle IHJ \): Given
3. \( \overline{HJ}\cong\overline{HJ} \): Reflexive property of congruence
4. \( \triangle HIJ\cong\triangle JKH \): ASA (Angle - Side - Angle) congruence criterion