Question

complete the proof that \\(\overline{jk} \cong \overline{ik}\\).

(image of triangle with points i, h, k, j and a table with statements and reasons)

statement | reason
--- | ---
1 \\(\angle i \cong \angle j\\) | given
2 \\(\overline{hk} \perp \overline{id}\\) | given
3 \\(\angle hki \cong \angle hkj\\) |
4 \\(\overline{hk} \cong \overline{hk}\\) |
5 \\(\triangle hik \cong \triangle hjk\\) |
6 \\(\overline{jk} \cong \overline{ik}\\) |

Explanation:

Step1: Definition of perpendicular lines

If \(\overline{HK}\perp\overline{IJ}\), then \(\angle HKI\) and \(\angle HKJ\) are right angles. By the definition of right angles, \(\angle HKI\cong\angle HKJ\) (all right angles are congruent).

Step2: Reflexive property of congruence

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

Step3: AAS (Angle - Angle - Side) congruence criterion

We know that \(\angle I\cong\angle J\) (given), \(\angle HKI\cong\angle HKJ\) (from step 1), and \(\overline{HK}\cong\overline{HK}\) (from step 2). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle HIK\cong\triangle HJK\).

Step4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle HIK\cong\triangle HJK\), their corresponding parts are congruent. So, \(\overline{JK}\cong\overline{IK}\) by CPCTC.

Answer:

1. Reason: Definition of perpendicular lines (all right angles are congruent)
2. Reason: Reflexive property of congruence
3. Reason: AAS (Angle - Angle - Side)
4. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)