Question

each diagram shows two triangles with two congruent angles or sides. identify one additional pair of corresponding angles or sides such that, if the pair were congruent, the two triangles could be proved congruent by aas.
7.
8.
9.
10.
11.
12.

Explanation:

Step1: Recall AAS (Angle - Angle - Side) congruence criterion

AAS states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step2: Analyze problem 9

In problem 9, we already have \(\angle B\cong\angle E\) and \(\angle C\cong\angle F\) (from the diagram's given congruent angles). For AAS, we need a non - included side.
The non - included sides are \(AB\) and \(DE\) (or \(AC\) and \(DF\)). So, if \(\overline{AB}\cong\overline{DE}\) (or \(\overline{AC}\cong\overline{DF}\)), by AAS, \(\triangle ABC\cong\triangle DEF\)

Step3: Analyze problem 10

In problem 10, we have \(\angle C\cong\angle F\) (from the diagram's marked congruent angles) and \(BC = EF\) (marked as equal). If \(\angle A\cong\angle D\), then by AAS (two angles \(\angle A\cong\angle D\), \(\angle C\cong\angle F\) and non - included side \(BC\cong EF\)), \(\triangle ABC\cong\triangle DEF\)

Step4: Analyze problem 11

In problem 11, we have \(\angle A\cong\angle E\) (from the diagram's marked congruent angles) and \(\angle BCA\cong\angle DCE\) (vertical angles are congruent). If \(\overline{AB}\cong\overline{DE}\), then by AAS (two angles \(\angle A\cong\angle E\), \(\angle BCA\cong\angle DCE\) and non - included side \(\overline{AB}\cong\overline{DE}\)), \(\triangle ABC\cong\triangle DEC\)

Answer:

1. \(\overline{AB}\cong\overline{DE}\), or \(\overline{BC}\cong\overline{EF}\)
2. \(\angle A\cong\angle D\)
3. \(\overline{AB}\cong\overline{DE}\)