Question

∠k ≅ ∠j. complete the proof that ̅{ij} ≅ ̅{ik}.

Explanation:

Step1: Identify the given information

We are given that \(\angle K\cong\angle J\).

Step2: Use the reflexive property

For any segment \(JK\), by the reflexive property of congruence, \(\overline{JK}\cong\overline{JK}\).

Step3: Apply the AAS (Angle - Angle - Side) congruence criterion

In \(\triangle IJK\) and \(\triangle IKJ\), we have \(\angle K\cong\angle J\) (given), \(\overline{JK}\cong\overline{JK}\) (reflexive property), and \(\angle I\) is common to both triangles. So, \(\triangle IJK\cong\triangle IKJ\) by AAS.

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

Since \(\triangle IJK\cong\triangle IKJ\), then \(\overline{IJ}\cong\overline{IK}\) (CPCTC).

Answer:

The reason for \(\angle K\cong\angle J\) is "Given". The reason for \(\overline{JK}\cong\overline{JK}\) is "Reflexive Property of Congruence". The reason for \(\triangle IJK\cong\triangle IKJ\) is "AAS". The reason for \(\overline{IJ}\cong\overline{IK}\) is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".