Question

Given: FG = TG, GH ⊥ FI

Prove: ∠AHF ≅ ∠AHG
A) Given
B) Definition of perpendicular lines
C) Substitution
D) Angle congruence

Answer 1

To prove ∠AHF ≅ ∠AHG, you use the given information that FG = TG and GH ⊥ FI, along with the definitions of perpendicular lines, substitution, and angle congruence to show that triangles FGH and TGH are congruent, which in turn proves the angle congruence.

Step-by-step explanation:

To prove that ∠AHF ≅ ∠AHG given that FG = TG and GH ⊥ FI, we can follow these steps:

1. Given: FG = TG and GH ⊥ FI.
   
2. Definition of perpendicular lines: Because GH is perpendicular to FI, we know that ∠GHF and ∠GHI are right angles and thus congruent.
   
3. Substitution: If FG = TG (given), and GH is a part of both ΔFGH and ΔTGH, by the Reflexive Property, side GH is congruent to itself in both triangles.
   
4. Angle congruence: With two angles and a non-included side congruent in ΔFGH and ΔTGH (angle-side-angle), the triangles are congruent by ASA Congruence Postulate. Consequently, ∠AHF ≅ ∠AHG by corresponding parts of congruent triangles (CPCTC).