Question

∠gjk ≅ ∠hji, ¯¯gk ⊥ ¯¯gj, ¯¯hj ⊥ ¯¯ji, and ¯¯gk ≅ ¯¯hj. complete the proof that ¯¯jk ≅ ¯¯ji

(there is a diagram of a geometric figure with points g, k, j, h, i)

statement	reasons
2. ¯¯gk ⊥ ¯¯gj
3. ¯¯hj ⊥ ¯¯ji
4. ¯¯gk ≅ ¯¯hj
5. ∠k ≅ ∠j
6. ∆gjk ≅ ∆hji
7. ¯¯jk ≅ ¯¯ji

Explanation:

Step1: Identify Given Information

We know \( \angle GJK \cong \angle HJI \) (given), \( \overline{GK} \perp \overline{KJ} \), \( \overline{HI} \perp \overline{JI} \) (so \( \angle GKJ = \angle HIJ = 90^\circ \), definition of perpendicular), and \( \overline{KJ} \cong \overline{JI} \) (given).

Step2: Prove Triangles Congruent

Triangles \( \triangle GJK \) and \( \triangle HIJ \) have:

• \( \angle GJK \cong \angle HJI \) (given)
• \( \overline{KJ} \cong \overline{JI} \) (given)
• \( \angle GKJ \cong \angle HIJ = 90^\circ \) (right angles)

By ASA (Angle - Side - Angle) congruence criterion, \( \triangle GJK \cong \triangle HIJ \).

Step3: Corresponding Parts of Congruent Triangles

Since \( \triangle GJK \cong \triangle HIJ \), their corresponding sides \( \overline{GK} \) and \( \overline{HI} \) are congruent (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Answer:

To complete the proof:

1. Reason for \( \angle GJK \cong \angle HJI \): Given
2. Reason for \( \overline{GK} \perp \overline{KJ} \), \( \overline{HI} \perp \overline{JI} \): Definition of perpendicular lines (forms \( 90^\circ \) angles)
3. Reason for \( \angle GKJ \cong \angle HIJ \): All right angles are congruent
4. Reason for \( \overline{KJ} \cong \overline{JI} \): Given
5. Reason for \( \triangle GJK \cong \triangle HIJ \): ASA (Angle - Side - Angle) Congruence Criterion ( \( \angle GJK \cong \angle HJI \), \( \overline{KJ} \cong \overline{JI} \), \( \angle GKJ \cong \angle HIJ \) )
6. Reason for \( \overline{GK} \cong \overline{HI} \): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

The key steps use ASA congruence and CPCTC to show \( \overline{GK} \cong \overline{HI} \).