Question

δhik ≅ δjgk. complete the proof that δijk ≅ δghk.
1 δhik ≅ δjgk given
2 overline { gk } ≅ overline { ik } cpctc
3 overline { hk } ≅ overline { jk } cpctc
4 ∠gkh ≅ ∠ikj vertical angle theorem
5 δijk ≅ δghk sas

Explanation:

Step1: Use CPCTC

Since \(\triangle HIK\cong\triangle JGK\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\overline{GK}\cong\overline{IK}\) and \(\overline{HK}\cong\overline{JK}\).

Step2: Use Vertical Angle Theorem

\(\angle GKH\) and \(\angle IKJ\) are vertical angles. By the Vertical Angle Theorem, \(\angle GKH\cong\angle IKJ\).

Step3: Use SAS Congruence

In \(\triangle IJK\) and \(\triangle GHK\), we have \(\overline{IK}\cong\overline{GK}\), \(\angle IKJ\cong\angle GKH\), and \(\overline{JK}\cong\overline{HK}\). By the SAS (Side - Angle - Side) Congruence Theorem, \(\triangle IJK\cong\triangle GHK\).

Answer:

The proof is completed as shown in the steps above.