Question

1. use the triangles at right to answer the following questions. are the triangles at right similar? a. how do you know if they are similar? show your reasoning in a flowchart. b. examine your work from part (a). are the triangles also congruent? explain why or why not.

Explanation:

Step1: Recall similarity criterion

Two triangles are similar if their corresponding angles are equal. In \(\triangle ABC\) and \(\triangle DEF\), \(\angle A = 74^{\circ}\), \(\angle D=74^{\circ}\), \(\angle B = 9^{\circ}\), \(\angle F = 9^{\circ}\). By the AA (angle - angle) similarity criterion, since two pairs of corresponding angles are equal, the triangles are similar.

Step2: Recall congruence criterion

Two triangles are congruent if their corresponding angles are equal and corresponding sides are equal. Given \(AC = 4\) and \(DF=4\), but we have no information about the other side - lengths. Just having two pairs of equal angles and one pair of equal non - included sides is not enough to prove congruence. We would need more information such as another pair of equal sides or the included side between the equal angles to be equal.

Answer:

a. The triangles are similar. We know this by the AA (angle - angle) similarity criterion since \(\angle A=\angle D = 74^{\circ}\) and \(\angle B=\angle F = 9^{\circ}\).
b. The triangles are not necessarily congruent. While they have two pairs of equal angles and one pair of equal sides (\(AC = DF = 4\)), we do not have enough information (such as another pair of equal sides or the included side between the equal angles being equal) to satisfy a congruence criterion like ASA, SAS, SSS, AAS, or HL.