Question

1. identify the missing information from the 2 - column proof. statements: hg ≅ ed, gi ≅ df, δhgi ≅ δedf. reasons: given, given, sas congruence postulate. multiple - choice options: m ≅ fe, gi ≅ fd, ∠g ≅ ∠d, ∠h ≅ ∠e. (diagram of two triangles with sides 5, 11 and angle 123°)

Explanation:

Step1: Analyze the SAS Congruence Postulate

The SAS (Side - Angle - Side) Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In the given proof, we have two sides already given as congruent (\(HG\cong ED\) and \(GI\cong DF\)). For the SAS postulate, we need the included angle between these two sides to be congruent. In \(\triangle HGI\) and \(\triangle EDF\), the included angle for \(HG\) and \(GI\) is \(\angle G\), and the included angle for \(ED\) and \(DF\) is \(\angle D\). So we need \(\angle G\cong\angle D\) to apply the SAS postulate for proving \(\triangle HGI\cong\triangle EDF\).

Step2: Evaluate the Options

• Option 1: \(HI\cong FE\) - This is a side, but we are using SAS, and we already have two sides. We need the included angle, not another side.
• Option 2: \(GI\cong FD\) - This is already given as a statement with the reason "Given", so it's not the missing information.
• Option 3: \(\angle G\cong\angle D\) - This is the included angle between the two given congruent sides, which is required for the SAS Congruence Postulate to prove the triangles congruent.
• Option 4: \(\angle H\cong\angle E\) - These are not the included angles between the given congruent sides, so this does not help in applying the SAS postulate.

Answer:

\(\boldsymbol{\angle G\cong\angle D}\) (corresponding to the option with \(\angle G\cong\angle D\) in the multiple - choice options)