Question

complete the proof that ( \triangle hjk cong \triangle hji ).

Explanation:

Step1: Identify the first reason

From the figure, we can see that the segments \( \overline{IJ} \) and \( \overline{JK} \) have the same number of tick - marks. So, the reason for \( \overline{IJ}\cong\overline{JK} \) is "Given (from the diagram's markings)".

Step2: Identify the second reason

The segments \( \overline{HK} \) and \( \overline{HI} \) have the same number of tick - marks. So, the reason for \( \overline{HK}\cong\overline{HI} \) is "Given (from the diagram's markings)".

Step3: Identify the third reason

The segment \( \overline{HJ} \) is common to both \( \triangle HJK \) and \( \triangle HJI \). So, the reason for \( \overline{HJ}\cong\overline{HJ} \) is "Reflexive property of congruence (\( a\cong a \) for any segment \( a \))".

Step4: Identify the fourth reason

We have three pairs of congruent sides (\( SSS - Side - Side - Side \) congruence criterion). So, the reason for \( \triangle HJK\cong\triangle HJI \) is "SSS (Side - Side - Side) congruence theorem".

Answer:

1. Given (from the diagram's markings)
2. Given (from the diagram's markings)
3. Reflexive property of congruence
4. SSS (Side - Side - Side) congruence theorem