Question

proving congruent triangles using transformations
determine the rigid transformations that will map
△abc to △xyz.
○ translate vertex b to vertex z; reflect △abc across
side ab.
○ translate vertex x to vertex c; rotate △xyz to align
the sides and angles.
○ reflect △abc across side ab; translate vertex c to
vertex x.
○ translate vertex x to vertex a; rotate △xyz to align
the sides and angles.

Explanation:

Step1: Analyze triangle angles

First, find the third angle of \( \triangle ABC \). The sum of angles in a triangle is \( 180^\circ \). So, \( \angle C = 180^\circ - 98^\circ - 35^\circ = 47^\circ \). In \( \triangle XYZ \), \( \angle Z = 47^\circ \), \( \angle X = 35^\circ \), so \( \angle Y = 180^\circ - 47^\circ - 35^\circ = 98^\circ \). Thus, \( \triangle ABC \) and \( \triangle XYZ \) have corresponding angles: \( \angle A = \angle X = 35^\circ \), \( \angle B = \angle Y = 98^\circ \), \( \angle C = \angle Z = 47^\circ \), and sides with the same tick marks are equal (so \( AC = XZ \) as they have one tick mark).

Step2: Evaluate each option

• Option 1: Translating B to Z and reflecting over AB. This doesn't align corresponding parts properly as B should correspond to Y (since \( \angle B = \angle Y = 98^\circ \)), not Z. So incorrect.
• Option 2: Translating X to C. But X corresponds to A ( \( \angle X = \angle A = 35^\circ \) ), so translating X to C is misaligning. Incorrect.
• Option 3: Reflecting over AB and translating C to X. The reflection and translation here don't match the correspondence (C corresponds to Z, not X). Incorrect.
• Option 4: Translate vertex X to vertex A (since \( \angle X = \angle A = 35^\circ \), they are corresponding vertices). Then rotating \( \triangle XYZ \) will align the sides and angles because after translating X to A, the angles and sides will match up (as \( AC = XZ \), and angles are equal), so rotating will align the other vertices (Y to B, Z to C). This works.

Answer:

Translate vertex X to vertex A; rotate \( \triangle XYZ \) to align the sides and angles. (The fourth option: "Translate vertex X to vertex A; rotate \( \triangle XYZ \) to align the sides and angles.")