Question

14 mark for review triangle abc and def are similar, where a corresponds to d, and b corresponds to e. the measure of angle a is 22°, and ab = 4. which statement(s) must be true ? i. the measure of angle d is 22° ii. de = 4 a i only b ii only c i and ii d neither i nor ii

Explanation:

Step1: Recall properties of similar triangles

Similar triangles have corresponding angles equal and corresponding sides in proportion (not necessarily equal in length unless they are congruent, a special case of similar triangles with scale factor 1).

Step2: Analyze Statement I

Since \( \triangle ABC \sim \triangle DEF \) and \( A \) corresponds to \( D \), corresponding angles are equal. So \( \angle A=\angle D \). Given \( \angle A = 22^\circ \), then \( \angle D=22^\circ \). So Statement I is true.

Step3: Analyze Statement II

Corresponding sides of similar triangles are in proportion, i.e., \( \frac{AB}{DE}=\text{scale factor} \). We only know \( AB = 4 \), but we don't know the scale factor. The scale factor could be 1 (making \( DE = 4 \)) or any other positive real number (making \( DE \) not equal to 4). So we cannot conclude \( DE = 4 \) must be true. Thus, Statement II is not necessarily true.

Answer:

A. I only