Question

use the given information to prove that \\( \triangle def \cong \triangle dgf \\).
given: \\( \angle dfe \\) is a right angle
\\( \angle dfg \\) is a right angle
\\( \angle edf \cong \angle gdf \\)
prove: \\( \triangle def \cong \triangle dgf \\)

statement	reason
2 \\( \angle dfg \\) is a right angle	given
3 \\( \angle dfe \cong \angle \\)	all right angles are \\( \cong \\)
4 \\( \angle edf \cong \angle gdf \\)	reason?
5 \\( \overline { df } \cong \overline { df } \\)	reason?
6 \\( \triangle def \cong \triangle dgf \\)	reason?

Explanation:

Step1: Identify the right - angle equality

Since \(\angle DFE\) and \(\angle DFG\) are both right angles, by the property that all right angles are congruent, \(\angle DFE\cong\angle DFG\).

Step2: Analyze \(\angle EDF\cong\angle GDF\)

This is given in the problem statement.

Step3: Analyze \(\overline{DF}\cong\overline{DF}\)

By the reflexive property of congruence (a segment is congruent to itself), \(\overline{DF}\cong\overline{DF}\).

Step4: Apply the Angle - Angle - Side (AAS) congruence criterion

In \(\triangle DEF\) and \(\triangle DGF\), we have \(\angle DFE\cong\angle DFG\) (from Step1), \(\angle EDF\cong\angle GDF\) (given), and \(\overline{DF}\cong\overline{DF}\) (from Step3). 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.

Answer:

1. \(DFG\)
2. Given
3. Reflexive property of congruence
4. AAS (Angle - Angle - Side) congruence criterion