Question

in the diagram below, ∠dfg ≅ ∠dgf and ∠edf ≅ ∠gdh.
complete the proof that (overline{de}congoverline{dh}).
in the diagram, ∠dfg ≅ ∠dgf and ∠edf ≅ ∠gdh. by the converse of the isosceles triangle theorem,
since ∠efd and ∠dfg form a linear pair, they are supplementary. so, for the same reason, ∠dgf and ∠dgh are supplementary. by the congruent supplements theorem,
. so, δdef ≅ δdhg by the
theorem. since
are congruent, (overline{de}congoverline{dh}).

Explanation:

Step1: Apply the Converse of the Isosceles Triangle Theorem

The Converse of the Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. Since \( \angle DFG\cong\angle DGF\), in \(\triangle DFG\), the sides opposite these angles are \(DF\) and \(DG\). So, \(DF = DG\).

Step2: Use the Congruent Supplements Theorem

If two angles are congruent, then their supplements are congruent. Given \( \angle DFG\cong\angle DGF\), and \( \angle EFD\) is supplementary to \( \angle DFG\) (\(\angle EFD+\angle DFG = 180^{\circ}\)), \( \angle DGH\) is supplementary to \( \angle DGF\) (\(\angle DGH+\angle DGF=180^{\circ}\)). By the Congruent Supplements Theorem, \( \angle EFD\cong\angle DGH\).

Step3: Prove triangle congruence

We know that \( \angle EDF\cong\angle GDH\) (given) and \(DF = DG\) (from Step 1). The triangles \(\triangle DEF\) and \(\triangle DHG\) have two pairs of congruent angles (\( \angle EDF\cong\angle GDH\) and \( \angle EFD\cong\angle DGH\)) and a pair of congruent sides (\(DF = DG\)). By the Angle - Side - Angle (ASA) Congruence Theorem, \(\triangle DEF\cong\triangle DHG\).

Step4: Conclude the side - congruence

Since \(\triangle DEF\cong\triangle DHG\), corresponding parts of congruent triangles are congruent. So, \(DE\) and \(DH\) (corresponding sides of \(\triangle DEF\) and \(\triangle DHG\)) are congruent.

Answer:

1. \(DF\cong DG\)
2. \( \angle EFD\cong\angle DGH\), ASA, corresponding sides (\(DE\) and \(DH\))