Question

gk ≅ hk and jk ≅ ik. complete the proof that ∠j ≅ ∠i.

1 gk ≅ hk
2 jk ≅ ik
3 ∠gki ≅ ∠hki
4 △gjk ≅ △hik
5 ∠j ≅ ∠i

Explanation:

Step1: Given

We are given that \(\overline{GK}\cong\overline{HK}\) and \(\overline{JK}\cong\overline{IK}\) as per the problem statement.

Step2: Vertical Angles Theorem

\(\angle GKJ\) and \(\angle HKI\) are vertical angles. By the Vertical Angles Theorem, \(\angle GKJ\cong\angle HKI\).

Step3: SAS (Side - Angle - Side) Congruence Criterion

In \(\triangle GJK\) and \(\triangle HIK\), we have \(\overline{GK}\cong\overline{HK}\), \(\angle GKJ\cong\angle HKI\), and \(\overline{JK}\cong\overline{IK}\). So, by the SAS congruence criterion, \(\triangle GJK\cong\triangle HIK\).

Step4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle GJK\cong\triangle HIK\), their corresponding angles \(\angle J\) and \(\angle I\) are congruent. That is, \(\angle J\cong\angle I\) by CPCTC.

Answer:

1. Given
2. Given
3. Vertical Angles Theorem
4. SAS (Side - Angle - Side)
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)