Question

→← gj bisects ∠hgi. complete the proof that hj ≅ ij.
triangle with vertices i, g, h; j on ih; gi and gh marked congruent; gj drawn

statement	reason
2. gi ≅ gh	given
3. ∠hgj ≅ ∠igj	definition of angle bisector
4. gj ≅ gj	reflexive property of congruence
5. △ghj ≅ △gij	dropdown
6. hj ≅ ij	dropdown

Explanation:

Step1: Identify congruent triangles

We have two sides and the included angle congruent between $\triangle GHJ$ and $\triangle GIJ$: $\overline{GI} \cong \overline{GH}$, $\angle HGJ \cong \angle IGJ$, and $\overline{GJ} \cong \overline{GJ}$. This matches the SAS (Side-Angle-Side) Congruence Postulate.

Step2: Corresponding parts of congruent triangles

If $\triangle GHJ \cong \triangle GIJ$, then their corresponding sides are congruent. $\overline{HJ}$ and $\overline{IJ}$ are corresponding sides of the congruent triangles, so they are congruent by the definition of congruent triangles (Corresponding Parts of Congruent Triangles are Congruent, CPCTC).

Answer:

Statement	Reason
2. $\overline{GI} \cong \overline{GH}$	Given
3. $\angle HGJ \cong \angle IGJ$	Definition of angle bisector
4. $\overline{GJ} \cong \overline{GJ}$	Reflexive Property of Congruence
5. $\triangle GHJ \cong \triangle GIJ$	SAS (Side-Angle-Side) Congruence Postulate
6. $\overline{HJ} \cong \overline{IJ}$	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)