Question

hi ≅ hj. complete the proof that △hij ≅ △hji.
1 hi ≅ hj
2 ∠h ≅ ∠h
3 △hij ≅ △hji

Explanation:

Step1: Given

The problem states that \(\overline{HI}\cong\overline{HJ}\), so the reason for statement 1 is "Given".

Step2: Reflexive property of congruence

For any angle \(\angle A\), \(\angle A\cong\angle A\). Here, \(\angle H\) is common to both \(\triangle HIJ\) and \(\triangle HJI\), so \(\angle H\cong\angle H\) by the reflexive property of congruence.

Step3: SAS (Side - Angle - Side) congruence criterion

The SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In \(\triangle HIJ\) and \(\triangle HJI\), we have \(\overline{HI}\cong\overline{HJ}\) (given), \(\angle H\cong\angle H\) (reflexive property), and \(\overline{HJ}\cong\overline{HI}\) (given, commutative property of congruence for sides). The included angle is \(\angle H\). So, \(\triangle HIJ\cong\triangle HJI\) by the SAS congruence criterion.

Answer:

1. Given
2. Reflexive property of congruence
3. SAS (Side - Angle - Side) congruence criterion