Question

shawana used a reflection followed by a dilation to transform △abc to show △abc as shown. which of the following statements is true? angle a is congruent to angle a angle b is 130 degrees segment ab is equal to segment ab triangle abc is congruent to triangle abc

Explanation:

Step1: Recall transformations

Reflection preserves angle measures and side lengths (congruence), dilation preserves angle measures but scales side lengths (similarity). So \(\triangle ABC \sim \triangle A'B'C'\), angles are congruent, sides are proportional.

Step2: Analyze each option

• Option 1: Angle \(A\) and \(A'\) are corresponding angles in similar triangles, so congruent. Let's check others.
• Option 2: In \(\triangle A'B'C'\), \(\angle B' = 110^\circ\), \(\angle C' = 45^\circ\), so \(\angle A' = 180 - 110 - 45 = 25^\circ\). Since \(\angle A=\angle A'\), \(\angle B = 180 - \angle A - \angle C\). But \(\angle C\) in \(\triangle ABC\) corresponds to \(\angle C' = 45^\circ\), so \(\angle B = 180 - 25 - 45 = 110^\circ\)? Wait, no—wait, reflection then dilation: reflection is congruence, dilation is similarity. Wait, maybe I messed up. Wait, reflection preserves angles, dilation also preserves angles (similarity). So \(\angle B\) should equal \(\angle B'\)? Wait no, the diagram: \(\triangle ABC\) and \(\triangle A'B'C'\), with \(\angle B' = 110^\circ\), \(\angle C' = 45^\circ\). So \(\angle A' = 25^\circ\). Then \(\angle A = \angle A'\), \(\angle B = \angle B'\)? Wait no, maybe the triangles are similar, so corresponding angles are equal. Wait, let's recalculate \(\angle A'\): \(180 - 110 - 45 = 25^\circ\). So \(\angle A = 25^\circ\), \(\angle B\) in \(\triangle ABC\) should be equal to \(\angle B'\) if corresponding? Wait, maybe the first option: Angle \(A\) congruent to \(A'\) is true because reflection and dilation both preserve angle measures (reflection is isometry, dilation is similarity transformation, which preserves angle measures).

• Option 2: If \(\angle B\) were 130, but in \(\triangle A'B'C'\), angles sum to 180, so \(\angle A' = 25\), so \(\angle A = 25\), \(\angle C = 45\) (corresponding to \(\angle C'\)), so \(\angle B = 180 - 25 - 45 = 110\), so option 2 is false.

• Option 3: Dilation scales side lengths, so \(AB\) and \(A'B'\) are proportional, not equal (unless scale factor 1, but since \(\triangle A'B'C'\) is larger, scale factor >1, so \(A'B' > AB\)), so false.

• Option 4: Congruent means same size and shape. Dilation changes size, so they are similar, not congruent. So false.

Answer:

Angle A is congruent to angle A'