Question

list the pairs of congruent angles. then write the extended proportion that related the corresponding sides for the similar polygons.
1 △xyz ~ △mnp
2 abcd ~ hjkl
tell whether the polygons are similar. if they are similar, write a similarity statement and give the scale factor.
3
4
5
6

Explanation:

Step1: Identify congruent - angles for similar polygons

For similar polygons, corresponding angles are congruent.
For $\triangle XYZ\sim\triangle MNP$, $\angle X\cong\angle M$, $\angle Y\cong\angle N$, $\angle Z\cong\angle P$. The extended - proportion of corresponding sides is $\frac{XY}{MN}=\frac{YZ}{NP}=\frac{XZ}{MP}$.
For $ABCD\sim HIKL$, $\angle A\cong\angle H$, $\angle B\cong\angle I$, $\angle C\cong\angle K$, $\angle D\cong\angle L$. The extended - proportion of corresponding sides is $\frac{AB}{HI}=\frac{BC}{IK}=\frac{CD}{KL}=\frac{DA}{LH}$.

Step2: Determine similarity and scale - factor

For polygons $WXYZ$ and $ABCD$:

The ratios of corresponding sides are $\frac{WX}{AB}=\frac{10}{15}=\frac{2}{3}$, $\frac{XY}{BC}=\frac{4}{6}=\frac{2}{3}$, $\frac{YZ}{CD}=\frac{10}{15}=\frac{2}{3}$, $\frac{ZW}{DA}=\frac{4}{6}=\frac{2}{3}$. Since all corresponding - side ratios are equal and corresponding angles are congruent (as they are parallelograms with equal - measure angles), $WXYZ\sim ABCD$ with a scale factor of $\frac{2}{3}$.

For polygons $DEFG$ and $RST$:

$\frac{DE}{RS}=\frac{16}{32}=\frac{1}{2}$, $\frac{EF}{ST}=\frac{34}{68}=\frac{1}{2}$, $\frac{DF}{RT}=\frac{30}{60}=\frac{1}{2}$. Since all corresponding - side ratios are equal and corresponding angles are congruent (right - angled triangles with equal non - right angles), $DEFG\sim RST$ with a scale factor of $\frac{1}{2}$.

For polygons $PQRS$ and $GHIJ$:

$\frac{PQ}{GH}=\frac{5}{10}=\frac{1}{2}$, $\frac{QR}{HI}=\frac{9}{8}
eq\frac{1}{2}$, $\frac{RS}{IJ}=\frac{5}{8}
eq\frac{1}{2}$, $\frac{SP}{JG}=\frac{15}{25}=\frac{3}{5}
eq\frac{1}{2}$. So, $PQRS$ and $GHIJ$ are not similar.

For polygons $ABCD$ and $LMNP$:

$\frac{AB}{LM}=\frac{18}{6} = 3$, $\frac{BC}{MN}=\frac{12}{4}=3$, $\frac{CD}{NP}=\frac{6}{2}=3$, $\frac{DA}{PL}=\frac{18}{6}=3$. Since all corresponding - side ratios are equal and corresponding angles are congruent, $ABCD\sim LMNP$ with a scale factor of $3$.

Answer:

1. Congruent angles: $\angle X\cong\angle M$, $\angle Y\cong\angle N$, $\angle Z\cong\angle P$; Extended proportion: $\frac{XY}{MN}=\frac{YZ}{NP}=\frac{XZ}{MP}$
2. Congruent angles: $\angle A\cong\angle H$, $\angle B\cong\angle I$, $\angle C\cong\angle K$, $\angle D\cong\angle L$; Extended proportion: $\frac{AB}{HI}=\frac{BC}{IK}=\frac{CD}{KL}=\frac{DA}{LH}$
3. Similar, $WXYZ\sim ABCD$, scale factor $\frac{2}{3}$
4. Similar, $DEFG\sim RST$, scale factor $\frac{1}{2}$
5. Not similar
6. Similar, $ABCD\sim LMNP$, scale factor $3$