Question

congruence theorems
given: ∠fhg and ∠kjg are right angles;
fh ≅ kj
prove: △fgh ≅ △kgj
because all right angles are congruent.

1. we know that ∠fgh ≅ ∠kgj

because they are vertical angles.

1. △fgh ≅ △kgj because

(image of triangles fgh and kgj with right angles at h and j, and fh ≅ kj marked)

Explanation:

Step1: Identify known congruences

We know $\angle FHG \cong \angle KJG$ (right angles, all right angles are congruent), $\angle FGH \cong \angle KGJ$ (vertical angles), and $\overline{FH} \cong \overline{KJ}$ (given).

Step2: Apply AAS congruence

In triangles $\triangle FGH$ and $\triangle KGJ$:

• $\angle FHG \cong \angle KJG$ (right angles)
• $\angle FGH \cong \angle KGJ$ (vertical angles)
• $\overline{FH} \cong \overline{KJ}$ (given)

This satisfies the Angle - Angle - Side (AAS) congruence criterion for triangles. So $\triangle FGH \cong \triangle KGJ$ by AAS.

Answer:

The triangles are congruent by the AAS (Angle - Angle - Side) congruence theorem.