Question

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Explanation:

Step1: Fill Similar Polygons definitions

1. Polygons with the same shape but different size
2. Polygons are similar if:

(1) Corresponding angles are congruent
(2) Corresponding sides are proportional

1. The ratio of corresponding sides is called the scale factor
2. If polygons are similar, then their perimeters (and corresponding heights/altitudes) are also proportional

Step2: Calculate scale factor (△ABC to △DEF)

Take ratio of corresponding sides:
$\frac{AB}{DE} = \frac{8}{12} = \frac{2}{3}$

Step3: Calculate scale factor (△DEF to △ABC)

Take reciprocal of previous ratio:
$\frac{DE}{AB} = \frac{12}{8} = \frac{3}{2}$

Step4: Calculate perimeter ratio (△DEF to △ABC)

Perimeter ratio equals scale factor of △DEF to △ABC:
$\frac{3}{2}$

Step5: Write similarity statement

Match corresponding angles (marked angles):
$\triangle ABC \sim \triangle EDF$
Symbol for Similar: $\sim$

Step6: Solve △JKL ~ △PMN

Angles:

$\angle J \cong \angle P$, $\angle K \cong \angle M$, $\angle L \cong \angle N$

Sides:

$\frac{JK}{PM} = \frac{KL}{MN} = \frac{LJ}{NP}$

Step7: Solve △XYZ ~ △RYS

Angles:

$\angle X \cong \angle R$, $\angle Y \cong \angle Y$, $\angle Z \cong \angle S$

Sides:

$\frac{XY}{RY} = \frac{YZ}{YS} = \frac{ZX}{SR}$

Answer:

Similar Polygons Notes:

• Polygons with the same shape but different size
• Polygons are similar if:

(1) Corresponding angles are congruent
(2) Corresponding sides are proportional

• The ratio of corresponding sides is called the scale factor
• If polygons are similar, then their perimeters are also proportional

Scale Factor:

1. Scale factor of $\triangle ABC$ to $\triangle DEF$: $\frac{2}{3}$
2. Scale factor of $\triangle DEF$ to $\triangle ABC$: $\frac{3}{2}$
3. Ratio of perimeter of $\triangle DEF$ to $\triangle ABC$: $\frac{3}{2}$

Similarity Statements:

• Symbol for Similar: $\sim$
• Similarity statement: $\triangle ABC \sim \triangle EDF$

1. $\triangle JKL \sim \triangle PMN$

Angles	Sides
$\angle K \cong \angle M$	$\frac{KL}{MN} = \frac{LJ}{NP}$
$\angle L \cong \angle N$	$\frac{LJ}{NP} = \frac{JK}{PM}$

2. $\triangle XYZ \sim \triangle RYS$

Angles	Sides
$\angle Y \cong \angle Y$	$\frac{YZ}{YS} = \frac{ZX}{SR}$
$\angle Z \cong \angle S$	$\frac{ZX}{SR} = \frac{XY}{RY}$