Question

complete the proof that ( \triangle hik cong \triangle hjk ).

Explanation:

Step1: Definition of perpendicular lines

Since \( \overline{HK}\perp\overline{IJ}\), by the definition of perpendicular lines, \( \angle HKI = 90^{\circ}\) and \( \angle HKJ=90^{\circ}\). So, \( \angle HKI\cong\angle HKJ\) (all right angles are congruent).

Step2: Reflexive property of congruence

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

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

In \( \triangle HIK\) and \( \triangle HJK\), we have \( \angle IHK\cong\angle JHK\) (given), \( \overline{HK}\cong\overline{HK}\) (reflexive property), and \( \angle HKI\cong\angle HKJ\) (right angles). By the ASA (Angle - Side - Angle) congruence criterion, \( \triangle HIK\cong\triangle HJK\).

Answer:

1. All right angles are congruent; 4. Reflexive property of congruence; 5. ASA (Angle - Side - Angle) congruence criterion.