Question

part one: for each pair of similar figures list 1) the corresponding congruent angles, 2) the proportional sides and 3) write a similarity statement
1.
2.

Explanation:

Step1: Identify congruent angles

For triangles $\triangle ABC$ and $\triangle DEF$, $\angle A=\angle D = 39^{\circ}$, $\angle C=\angle F=81^{\circ}$, and since the sum of angles in a triangle is $180^{\circ}$, $\angle B=\angle E=180-(39 + 81)=60^{\circ}$.

Step2: Identify proportional sides

$\frac{AB}{DE}=\frac{24}{18}=\frac{4}{3}$, $\frac{BC}{EF}=\frac{20}{15}=\frac{4}{3}$, $\frac{AC}{DF}=\frac{12}{9}=\frac{4}{3}$

Step3: Write similarity statement

$\triangle ABC\sim\triangle DEF$

For rhombuses $ABCD$ and $EFGH$:

Step1: Identify congruent angles

Opposite - angles of a rhombus are equal. In rhombus $ABCD$, $\angle A=\angle C$, $\angle B=\angle D$, and in rhombus $EFGH$, $\angle E=\angle G$, $\angle F=\angle H$. Also, since the ratio of side - lengths $\frac{AB}{EF}=\frac{15}{7.5} = 2$, and corresponding angles are equal because the shape is similar.

Step2: Identify proportional sides

$\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{DA}{HE}=2$

Step3: Write similarity statement

$ABCD\sim EFGH$

Answer:

1.

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

2.

• Corresponding congruent angles: $\angle A=\angle E$, $\angle B=\angle F$, $\angle C=\angle G$, $\angle D=\angle H$
• Proportional sides: $\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{DA}{HE}=2$
• Similarity statement: $ABCD\sim EFGH$