Questions: Given: angle DEG is congruent to angle FEG Prove: line segment DG is congruent to line segment FG

Transcript text: Given: $\angle D E G \cong \angle F E G$ Prove: $\overline{D G} \cong \overline{F G}$

Solution

Step 1: Identify givens

We are given that ∠DEG ≅ ∠FEG and EG ⊥ DF.

Step 2: Consider implications of EG ⊥ DF

EG ⊥ DF implies that angles ∠DGE and ∠FGE are right angles, and therefore congruent.

Step 3: Identify common side

Side EG is common to both triangles ΔDEG and ΔFEG.

Step 4: Apply Angle-Side-Angle congruence

Since ∠DEG ≅ ∠FEG, EG is a shared side, and ∠DGE ≅ ∠FGE, by the Angle-Side-Angle postulate, triangles ΔDEG and ΔFEG are congruent.

Step 5: Deduce congruence of corresponding sides

Since triangles ΔDEG and ΔFEG are congruent, their corresponding sides must also be congruent. This implies that DG ≅ FG.

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