Question

k is the midpoint of \\(\overline{gi}\\) and \\(\overline{hj}\\). complete the proof that \\(\angle h \cong \angle j\\).

	statement	reason
2	k is the midpoint of \\(\overline{hj}\\)	given
3	\\(\overline{gk} \cong \overline{ik}\\)	definition of midpoint
4	\\(\overline{hk} \cong \overline{jk}\\)
5	\\(\angle gkj \cong \angle hki\\)	vertical angle theorem
6	\\(\triangle gjk \cong \triangle ihk\\)
7	\\(\angle h \cong \angle j\\)

Explanation:

Step1: Analyze Statement 4

Since \( K \) is the midpoint of \( \overline{HJ} \), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So the reason for \( \overline{HK} \cong \overline{JK} \) is the Definition of midpoint.

Step2: Analyze Statement 6

We have \( \overline{GK} \cong \overline{IK} \) (from step 3), \( \overline{HK} \cong \overline{JK} \) (from step 4), and \( \angle GKJ \cong \angle HKI \) (vertical angles). By the Side - Angle - Side (SAS) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. So \( \triangle GJK \cong \triangle IHK \) by SAS.

Step3: Analyze Statement 7

Since \( \triangle GJK \cong \triangle IHK \), corresponding parts of congruent triangles are congruent (CPCTC). So \( \angle H \cong \angle J \) because they are corresponding angles of congruent triangles.

Filling in the blanks:

• For Statement 4, Reason: Definition of midpoint.
• For Statement 6, Reason: SAS (Side - Angle - Side) Congruence Postulate.
• For Statement 7, Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

• Statement 4 Reason: Definition of midpoint
• Statement 6 Reason: SAS (Side - Angle - Side) Congruence Postulate
• Statement 7 Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)