Question

name: savannah.tt
date:
similar figures and polygons - assignment
part one: for each pair of similar figures list 1) the corresponding congruent angles, 2) the proportional sides and 3) write a similarity statement

Explanation:

Step1: Identify corresponding congruent angles

In \(\triangle ABC\) and \(\triangle DEF\), \(\angle A=\angle D = 39^{\circ}\), \(\angle C=\angle F=81^{\circ}\), and \(\angle B=\angle E\) (since the sum of angles in a triangle is \(180^{\circ}\)). In the rhombuses \(ABCD\) and \(EFGH\), corresponding angles are congruent as they are similar polygons.

Step2: Find proportional sides

For \(\triangle ABC\) and \(\triangle DEF\), \(\frac{AB}{DE}=\frac{24}{18}=\frac{4}{3}\), \(\frac{BC}{EF}=\frac{20}{15}=\frac{4}{3}\), \(\frac{AC}{DF}=\frac{12}{9}=\frac{4}{3}\). For the rhombuses, \(\frac{AB}{EF}=\frac{15}{7.5} = 2\), \(\frac{BC}{FG}=\frac{15}{7.5}=2\), \(\frac{CD}{GH}=\frac{15}{7.5}=2\), \(\frac{DA}{HE}=\frac{15}{7.5}=2\).

Step3: Write similarity statements

For the triangles, \(\triangle ABC\sim\triangle DEF\). For the rhombuses, \(ABCD\sim EFGH\).

Answer:

For the triangles:

1. Corresponding congruent angles: \(\angle A\) and \(\angle D\), \(\angle C\) and \(\angle F\), \(\angle B\) and \(\angle E\).
2. Proportional sides: \(\frac{AB}{DE}=\frac{4}{3}\), \(\frac{BC}{EF}=\frac{4}{3}\), \(\frac{AC}{DF}=\frac{4}{3}\).
3. Similarity statement: \(\triangle ABC\sim\triangle DEF\).

For the rhombuses:

1. Corresponding congruent angles: All corresponding angles are congruent.
2. Proportional sides: \(\frac{AB}{EF} = 2\), \(\frac{BC}{FG}=2\), \(\frac{CD}{GH}=2\), \(\frac{DA}{HE}=2\).
3. Similarity statement: \(ABCD\sim EFGH\)