Question

1 j is the midpoint of \\( \overline { h i } \\)
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: Analyze each statement - reason pair

• For statement 1: \(J\) is the mid - point of \(\overline{HI}\). By the definition of a mid - point, if \(J\) is the mid - point of \(\overline{HI}\), then \(\overline{HJ}\cong\overline{IJ}\). So for statement 4, the reason is "Definition of midpoint".
• For statement 2: \(\overline{GJ}\perp\overline{HI}\). When two lines are perpendicular, the angles formed are right angles. So \(\angle GJH\) and \(\angle GJI\) are right angles. And all right angles are congruent. So for statement 3, the reason is "All right angles are congruent".
• For statement 5: \(\overline{GJ}\cong\overline{GJ}\). This is an example of the reflexive property of congruence (a segment is congruent to itself). So the reason for statement 5 is "Reflexive Property of Congruence".
• For statement 6: \(\triangle GHJ\cong\triangle GIJ\). We have \(\overline{HJ}\cong\overline{IJ}\) (from statement 4, definition of mid - point), \(\angle GJH\cong\angle GJI\) (from statement 3, all right angles are congruent), and \(\overline{GJ}\cong\overline{GJ}\) (from statement 5, reflexive property of congruence). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle GHJ\cong\triangle GIJ\). So the reason for statement 6 is "SAS".
• For statement 7: \(\angle HGJ\cong\angle IGJ\). Since \(\triangle GHJ\cong\triangle GIJ\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle HGJ\cong\angle IGJ\).

Answer:

1. Reason for statement 4: Definition of midpoint
2. Reason for statement 3: All right angles are congruent
3. Reason for statement 5: Reflexive Property of Congruence
4. Reason for statement 6: SAS
5. Reason for statement 7: CPCTC