Question

proof complete the paragraph proof given: ∠d ≅ ∠g, f is the midpoint of (overline{dg}). prove: ( \triangle def cong \triangle ghf ) proof: we are given that ( angle d cong angle g ). we are also given that f is the midpoint of (overline{dg}), so segment df is congruent to segment select choice by the definition of midpoint. we know that ( angle dfe cong angle gfh ) by the select choice theorem. from this, we know that ( \triangle def cong \triangle ghf ) by the select choice congruence postulates

Explanation:

Step1: Analyze Midpoint Property

Since \( F \) is the midpoint of \( \overline{DG} \), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So, \( \overline{DF} \cong \overline{GF} \).

Step2: Analyze Vertical Angles

\( \angle DFE \) and \( \angle GFH \) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, \( \angle DFE \cong \angle GFH \).

Step3: Determine Congruence Postulate

We are given \( \angle D \cong \angle G \), we have \( \overline{DF} \cong \overline{GF} \) (from midpoint) and \( \angle DFE \cong \angle GFH \) (vertical angles). So, by the Angle - Side - Angle (ASA) Congruence Postulate, \( \triangle DEF \cong \triangle GHF \).

Answer:

• First "Select Choice": ASA (Angle - Side - Angle)
• Second "Select Choice": \( \overline{GF} \)
• Third "Select Choice": Vertical Angles