Question

question 8 of 10
leon drew △abc and △def so that ∠a ≅ ∠d, ∠b ≅ ∠e, ab = 4, and
de = 8. are △abc and △def similar? if so, identify the similarity
postulate or theorem that applies.

a. similar - sss

b. cannot be determined

c. similar - sas

d. similar - aa

Explanation:

Step1: Recall similarity postulates

AA (Angle - Angle) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. SAS (Side - Angle - Side) similarity requires two sides in proportion and the included angle congruent. SSS (Side - Side - Side) similarity requires all three sides in proportion.

Step2: Analyze given information

We are given that \(\angle A\cong\angle D\) and \(\angle B\cong\angle E\). In any triangle, the sum of interior angles is \(180^{\circ}\), so if two angles of one triangle are congruent to two angles of another triangle, the third angles will also be congruent (since \(\angle C=180^{\circ}-\angle A - \angle B\) and \(\angle F = 180^{\circ}-\angle D-\angle E\), and \(\angle A=\angle D,\angle B = \angle E\) implies \(\angle C=\angle F\)). So, by AA similarity postulate, the triangles are similar. The sides \(AB = 4\) and \(DE=8\) are corresponding sides (since \(\angle A\) and \(\angle D\) are congruent, \(\angle B\) and \(\angle E\) are congruent, so \(AB\) corresponds to \(DE\)), but we don't need the side lengths for AA similarity. The SSS postulate is not applicable here as we don't have information about all three sides, and SAS is not applicable as we don't have two sides in proportion and the included angle (we have two angles and one side, not two sides and the included angle).

Answer:

D. Similar - AA