Question

gh ⊥ fg, ei ≅ fg, ∠ehi ≅ ∠fhg, and ei ⊥ hi. complete the proof that △ehi ≅ △fhg.

1 gh ⊥ fg given
2 ei ≅ fg given
3 ∠ehi ≅ ∠fhg given
4 ei ⊥ hi given
5 ∠i ≅ ∠g
6 △ehi ≅ △fhg aas additive property of angle measure additive property of length all right angles are congruent alternate interior angles theorem angles forming a linear pair sum to 180° asa corresponding angles theorem cpctc definition of angle bisector definition of congruence

Explanation:

Step1: Recall the property of right angles

Since \( \overline{GH}\perp\overline{FG}\), then \( \angle G = 90^{\circ}\) (by the definition of perpendicular lines: if two lines are perpendicular, the angle between them is \(90^{\circ}\)). Also, since \( \overline{EI}\perp\overline{HI}\), then \( \angle I=90^{\circ}\) (by the definition of perpendicular lines).

Step2: Use the congruence of angles

If \( \angle I = 90^{\circ}\) and \( \angle G=90^{\circ}\), then \( \angle I\cong\angle G\) (because all right - angles (\(90^{\circ}\) angles) are congruent).

Answer:

All right angles are congruent