Question

name
homework 8
date
directions: given the following information, complete the proofs below.

1. given: \\( \overline { e f } \cong \overline { g f } \\)

\\( \overline { e g } \perp \overline { h f } \\)
prove: \\( \triangle h f g \cong \triangle h f e \\)
describe the single rigid motion that maps one triangle onto the other.

Explanation:

Step1: Identify right angles

Since \(EG\perp HF\), \(\angle EHF = \angle GHF=90^{\circ}\) (Definition of perpendicular lines).

Step2: Use Hypotenuse - Leg (HL) criterion

In right - triangles \(\triangle HFE\) and \(\triangle HFG\):

• We are given that \(EF = GF\) (hypotenuse)
• \(HF\) is a common side (leg)

By the Hypotenuse - Leg (HL) congruence criterion for right - triangles (\(HL\): If the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent), \(\triangle HFE\cong\triangle HFG\)

Answer:

\(\triangle HFG\cong\triangle HFE\) by the Hypotenuse - Leg (\(HL\)) congruence criterion. The rigid motion that maps \(\triangle HFE\) onto \(\triangle HFG\) is a reflection over the line \(HF\) (since the two right - triangles are symmetric with respect to the line \(HF\)).