Question

proof complete the paragraph proof.
given: ∠d ≅ ∠g, f is the midpoint of overline{dg}.
prove: △def ≅ △ghf
proof: we are given that ∠d ≅ ∠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 ∠dfe ≅ ∠gfh by the
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theorem. from this, we know that △def ≅ △ghf by the
select choice
congruence postulate.

Explanation:

Step1: Mid - point property

If \(F\) is the mid - point of \(\overline{DG}\), then by the definition of mid - point, \(DF = FG\) (i.e., \(\overline{DF}\cong\overline{FG}\)).

Step2: Vertical angles

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

Step3: ASA congruence

We are given \(\angle D\cong\angle G\), we have \(\overline{DF}\cong\overline{FG}\) (from mid - point property) and \(\angle DFE\cong\angle GFH\) (from vertical angles). By the Angle - Side - Angle (ASA) Congruence Postulate, \(\triangle DEF\cong\triangle GHF\).

Answer:

First blank: \(\overline{DF}\cong\overline{FG}\), second blank: Vertical Angles, third blank: ASA