Question

in triangle abc and triangle def, ab and de are each equal to 5 centimeters, and angles a and d each have measure 35°. which additional piece of information is sufficient to prove that triangle abc is congruent to triangle def?
a the lengths of sides bc and ef are equal.
b the lengths of sides ac and df are equal.
c the measures of angles b and c are equal.
d no additional information is necessary to prove that the two triangles are congruent.

Explanation:

Step1: Recall congruence theorems

We know the Side - Angle - Side (SAS) congruence theorem. For two triangles \(\triangle ABC\) and \(\triangle DEF\), if \(AB = DE\), \(\angle A=\angle D\), and \(AC = DF\), then \(\triangle ABC\cong\triangle DEF\) by SAS.

Step2: Analyze each option

• Option A: If \(BC = EF\), we have \(AB = DE\), \(\angle A=\angle D\), and \(BC = EF\). This is the SSA (Side - Side - Angle) case, which is not a valid congruence theorem.
• Option B: If \(AC = DF\), since \(AB = DE = 5\) cm and \(\angle A=\angle D = 35^{\circ}\), by the SAS (Side - Angle - Side) congruence theorem, \(\triangle ABC\cong\triangle DEF\).
• Option C: If \(\angle B=\angle C\), this only gives information about the angles within \(\triangle ABC\) and does not help in proving congruence with \(\triangle DEF\).
• Option D: We already know \(AB = DE\) and \(\angle A=\angle D\). But this is not enough for congruence. We need another side or angle.

Answer:

B. The lengths of sides \(AC\) and \(DF\) are equal.