Question

17 mark for review in triangles abc and def, angles b and e each have measure 27° and angles c and f each have measure 41°. which additional piece of information is sufficient to determine whether triangle abc is congruent to triangle def? a the measure of angle a b the length of side ab c the lengths of sides bc and ef d no additional information is necessary.

Explanation:

Step1: Recall Triangle Congruence Rules

Triangle congruence rules (ASA, SAS, SSS, AAS, HL) require at least one pair of corresponding sides to be equal, along with angle conditions. Here, we know two angles in each triangle, so the third angle is fixed (since the sum of angles in a triangle is \(180^\circ\)). But angle - only information (AAA) doesn't prove congruence, it proves similarity.

Step2: Analyze Option A

The measure of angle \(A\): We already can calculate angle \(A\) (since \(m\angle A=180 - 27 - 41=112^\circ\)) and the same for angle \(D\) in \(\triangle DEF\). Just knowing the angle measure (which we can already find) doesn't give us side - length information, so it's not sufficient for congruence.

Step3: Analyze Option B

The length of side \(AB\): For congruence, we need a corresponding side. In \(\triangle ABC\) and \(\triangle DEF\), angle \(B\) corresponds to angle \(E\), angle \(C\) corresponds to angle \(F\). Side \(AB\) is adjacent to angle \(B\) and angle \(A\), but we don't know the corresponding side in \(\triangle DEF\) (like side \(DE\) or \(DF\) or \(EF\)) from this information alone. So this is not sufficient.

Step4: Analyze Option C

The lengths of sides \(BC\) and \(EF\): If \(BC = EF\), and we know that \(\angle B=\angle E = 27^\circ\) and \(\angle C=\angle F = 41^\circ\), then by the ASA (Angle - Side - Angle) congruence criterion (since we have two angles and the included side: \(\angle B\), side \(BC\), \(\angle C\) in \(\triangle ABC\) and \(\angle E\), side \(EF\), \(\angle F\) in \(\triangle DEF\)), we can conclude that \(\triangle ABC\cong\triangle DEF\).

Step5: Analyze Option D

No additional information is necessary: Since we only know angle measures, the triangles could be similar but not congruent (different side lengths). So this is incorrect.

Answer:

C. The lengths of sides \(BC\) and \(EF\)