Question

k is the midpoint of (overline{ij}) and (overline{hk} perp overline{ij}). complete the proof that (overline{hi} cong overline{hj}).

statement	reason
2 (overline{hk} perp overline{ij})	given
3 ( angle hki cong angle hkj )	all right angles are congruent
4 (overline{ik} cong overline{jk})	definition of midpoint
5 (overline{hk} cong overline{hk})	reflexive property of congruence
6

Explanation:

Step1: Identify Triangle Congruence

We have two right triangles \( \triangle HKI \) and \( \triangle HKJ \). \( K \) is the midpoint of \( \overline{IJ} \), so \( \overline{IK} \cong \overline{JK} \) (definition of midpoint). \( \overline{HK} \perp \overline{IJ} \), so \( \angle HKI \) and \( \angle HKJ \) are right angles (all right angles are congruent, so \( \angle HKI \cong \angle HKJ \)). \( \overline{HK} \cong \overline{HK} \) (reflexive property). By SAS (Side - Angle - Side) congruence criterion, \( \triangle HKI \cong \triangle HKJ \).

Step2: Corresponding Parts of Congruent Triangles

Since \( \triangle HKI \cong \triangle HKJ \), their corresponding sides \( \overline{HI} \) and \( \overline{HJ} \) are congruent. So the statement for step 6 should be \( \overline{HI} \cong \overline{HJ} \) and the reason is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Answer:

Statement 6: \( \overline{HI} \cong \overline{HJ} \)
Reason 6: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)