Question

△abc can be mapped onto △def using a series of dilations and rigid motions. which postulate or theorem proves that △abc is similar to △def? (and the image of two triangles with angles labeled: △abc has angles at a: 64°, at b: 61°; △def has angles at d: 64°, at e: 61°)

Explanation:

Step1: Analyze triangle angles

In $\triangle ABC$, $\angle A = 64^\circ$, $\angle B = 61^\circ$. So $\angle C = 180^\circ - 64^\circ - 61^\circ = 55^\circ$. In $\triangle DEF$, $\angle D = 64^\circ$, $\angle E = 61^\circ$, so $\angle F = 180^\circ - 64^\circ - 61^\circ = 55^\circ$. Thus, $\angle A=\angle D$, $\angle B=\angle E$, $\angle C=\angle F$.

Step2: Apply similarity postulate

The 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. Here, two pairs of angles are congruent ($\angle A\cong\angle D$, $\angle B\cong\angle E$), so by AA similarity, $\triangle ABC\sim\triangle DEF$. Also, the problem states a series of dilations and rigid motions map $\triangle ABC$ to $\triangle DEF$. Dilations preserve angle measures and create proportional sides, rigid motions preserve shape and size. The AA postulate is the key here as we have two congruent angles.

Answer:

The AA (Angle - Angle) similarity postulate proves $\triangle ABC$ is similar to $\triangle DEF$.