Question

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complete the proof that ( overline{jk} cong overline{ik} ).
1 ( angle jhk cong angle ihk )
2 ( angle j cong angle i )
3 ( overline{hk} cong overline{hk} )
4 ( \triangle hik cong \triangle hjk )
5 ( overline{jk} cong overline{ik} )

Explanation:

Step1: Given

The first statement \(\angle JHK\cong\angle IHK\) is given in the problem context.

Step2: Given

The second statement \(\angle J\cong\angle I\) is also given in the problem context.

Step3: Reflexive property of congruence

For any segment \(AB\), \(AB\cong AB\). So, for segment \(\overline{HK}\), \(\overline{HK}\cong\overline{HK}\) by the reflexive property of congruence.

Step4: AAS (Angle - Angle - Side) congruence criterion

If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. In \(\triangle HJK\) and \(\triangle HIK\), we have \(\angle J\cong\angle I\), \(\angle JHK\cong\angle IHK\), and \(\overline{HK}\cong\overline{HK}\). So, \(\triangle HIK\cong\triangle HJK\) by AAS.

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle HIK\cong\triangle HJK\), their corresponding parts \(\overline{JK}\) and \(\overline{IK}\) are congruent. So, \(\overline{JK}\cong\overline{IK}\) by CPCTC.

Answer:

1. Given
2. Given
3. Reflexive property of congruence
4. AAS (Angle - Angle - Side)
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)