Question

8 what additional information would be needed to prove these triangles are congruent by asa? a \\( \overline { k w } \cong \overline { b c } \\) b \\( \overline { k t } \cong \overline { a c } \\) c \\( \angle b \cong \angle w \\) d \\( \angle c \cong \angle k \\) e none of the above.

Explanation:

Step1: Recall ASA Congruence Criterion

ASA (Angle - Side - Angle) 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 Given Information

In \(\triangle ABC\) and \(\triangle WKT\), assume we have one pair of angles (\(\angle A\) and \(\angle T\)) and one pair of sides (the marked sides). For ASA, we need another pair of angles such that the side is included.

Step3: Check Each Option

• Option A: \(\overline{KW}\cong\overline{BC}\) gives a side - not the required angle for ASA.
• Option B: \(\overline{KT}\cong\overline{AC}\) gives a side - not the required angle for ASA.
• Option C: \(\angle B\cong\angle W\) gives the second pair of angles. If \(\angle A\cong\angle T\) (given by the arc markings), the marked sides (say \(AB = WT\) as per the single - tick markings), and \(\angle B\cong\angle W\), then by ASA (\(\angle A - AB-\angle B\) in \(\triangle ABC\) and \(\angle T - WT-\angle W\) in \(\triangle WKT\)), \(\triangle ABC\cong\triangle WKT\).
• Option D: \(\angle C\cong\angle K\) does not form the ASA pattern with the given side and angle.

Answer:

C. \(\angle B\cong\angle W\)