Question

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

1 gh ⊥ fg
2 ei ≅ fg
3 ∠ehi ≅ ∠fhg
4 ei ⊥ hi
5 ∠i ≅ ∠g
6 △ehi ≅ △fhg

Explanation:

Step1: Definition of perpendicular lines

If two lines are perpendicular, the angle between them is \(90^{\circ}\). So, since \(\overline{GH}\perp\overline{FG}\), \(\angle G = 90^{\circ}\). And since \(\overline{EI}\perp\overline{HI}\), \(\angle I=90^{\circ}\). By the definition of congruent angles (\(90^{\circ}=90^{\circ}\)), \(\angle I\cong\angle G\).

Step2: Angle - Angle - Side (AAS) congruence criterion

The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
We know that \(\angle EHI\cong\angle FHG\) (given), \(\angle I\cong\angle G\) (from Step 1), and \(\overline{EI}\cong\overline{FG}\) (given).

Answer:

1. Given
2. Given
3. Given
4. Given
5. Definition of perpendicular lines (since \(\overline{GH}\perp\overline{FG}\) and \(\overline{EI}\perp\overline{HI}\), \(\angle G = 90^{\circ}\) and \(\angle I = 90^{\circ}\), so \(\angle I\cong\angle G\))
6. AAS (Angle - Angle - Side) congruence criterion