Question

j is the midpoint of \\( \overline { h i } \\). complete the proof that \\( \angle h g j \cong \angle i g j \\).

statement	reason
2	\\( \overline { g j } \perp \overline { h i } \\)
3	\\( \angle g j h \cong \angle g j i \\)
4	\\( \overline { h j } \cong \overline { i j } \\)
5	\\( \overline { g j } \cong \overline { g j } \\)
6	\\( \triangle g h j \cong \triangle g i j \\)
7	\\( \angle h g j \cong \angle i g j \\)

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Given

Statement 2 is given in the problem.

Step3: Right Angle Congruence Theorem

Since \( \overline{GJ}\perp\overline{HI}\), \( \angle GJH = 90^{\circ}\) and \( \angle GJI=90^{\circ}\). By the Right - Angle Congruence Theorem, \( \angle GJH\cong\angle GJI\).

Step4: Definition of Midpoint

Since \( J\) is the midpoint of \( \overline{HI}\), by the definition of a midpoint, \( \overline{HJ}\cong\overline{IJ}\).

Step5: Reflexive Property of Congruence

For any segment \( \overline{AB}\), \( \overline{AB}\cong\overline{AB}\). So, \( \overline{GJ}\cong\overline{GJ}\) by the reflexive property of congruence.

Step6: SAS (Side - Angle - Side) Congruence Postulate

We have \( \overline{HJ}\cong\overline{IJ}\), \( \angle GJH\cong\angle GJI\), and \( \overline{GJ}\cong\overline{GJ}\). By the SAS (Side - Angle - Side) congruence postulate, \( \triangle GHJ\cong\triangle GIJ\).

Step7: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle GHJ\cong\triangle GIJ\), their corresponding parts are congruent. So, \( \angle HGJ\cong\angle IGJ\) by CPCTC.

Answer:

1. Given
2. Given
3. Right Angle Congruence Theorem
4. Definition of Midpoint
5. Reflexive Property of Congruence
6. SAS (Side - Angle - Side) Congruence Postulate
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)