Question

select the correct answer.
if $\triangle acb$ and $\triangle egf$ are congruent by the asa criterion, which pair of angles must be congruent?
a. $\angle acb$ and $\angle feg$
b. $\angle cab$ and $\angle gef$
c. $\angle cab$ and $\angle egf$
d. $\angle abc$ and $\angle gfe$

Explanation:

Step1: Recall ASA Criterion

ASA (Angle - Side - Angle) criterion for congruent triangles states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Step2: Analyze the Given Triangles

We know that \(\triangle ACB\cong\triangle EGF\) by ASA. Let's identify the corresponding parts. From the diagram, we can see the marked sides and angles. For ASA, we need two angles and the included side. Let's match the vertices: \(A\) corresponds to \(E\), \(C\) corresponds to \(G\), \(B\) corresponds to \(F\) (since \(\triangle ACB\) and \(\triangle EGF\) are congruent).

Now, let's check the angles:

• Option A: \(\angle ACB\) and \(\angle FEG\) - \(\angle ACB\) is at \(C\) in \(\triangle ACB\), \(\angle FEG\) is at \(E\) in \(\triangle EGF\). These are not corresponding angles for ASA.
• Option B: \(\angle CAB\) (at \(A\) in \(\triangle ACB\)) and \(\angle GEF\) (at \(E\) in \(\triangle EGF\)). Since \(A\) corresponds to \(E\), and for ASA, the angle at \(A\) ( \(\angle CAB\)) should correspond to the angle at \(E\) ( \(\angle GEF\)) when considering the included side.
• Option C: \(\angle CAB\) and \(\angle EGF\) - \(\angle EGF\) is at \(G\) in \(\triangle EGF\), which does not correspond to \(\angle CAB\) for ASA.
• Option D: \(\angle ABC\) and \(\angle GFE\) - These are angles at \(B\) and \(F\) respectively, but for ASA, we need the angle that is part of the two - angle - included - side combination. And from the correspondence \(A - E\), \(C - G\), \(B - F\), \(\angle ABC\) and \(\angle GFE\) are not the angles required for ASA (they are the non - included angles in the ASA sense here).

Answer:

B. \(\angle CAB\) and \(\angle GEF\)