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: Recall similarity rules

For similar polygons, corresponding angles are congruent and corresponding - side lengths are in proportion.

Step2: Problem 1

For $\triangle XYZ\sim\triangle MNP$, the pairs of congruent angles are $\angle X\cong\angle M$, $\angle Y\cong\angle N$, $\angle Z\cong\angle P$. The extended proportion of corresponding sides is $\frac{XY}{MN}=\frac{YZ}{NP}=\frac{XZ}{MP}$.

Step3: Problem 2

For $ABCD\sim HJKL$, the pairs of congruent angles are $\angle A\cong\angle H$, $\angle B\cong\angle J$, $\angle C\cong\angle K$, $\angle D\cong\angle L$. The extended proportion of corresponding sides is $\frac{AB}{HJ}=\frac{BC}{JK}=\frac{CD}{KL}=\frac{DA}{LH}$.

Step4: Problem 3

For quadrilaterals $WXYZ$ and $ABCD$:

• Check the ratios of corresponding sides. $\frac{WX}{AB}=\frac{10}{15}=\frac{2}{3}$, $\frac{XY}{BC}=\frac{4}{6}=\frac{2}{3}$, $\frac{YZ}{CD}=\frac{10}{15}=\frac{2}{3}$, $\frac{ZW}{DA}=\frac{4}{6}=\frac{2}{3}$.
• Since corresponding - side lengths are in proportion and we assume corresponding angles are congruent (if not given otherwise in a similar - polygon problem), the polygons are similar. The similarity statement is $WXYZ\sim ABCD$ and the scale factor is $\frac{2}{3}$.

Step5: Problem 4

For $\triangle DEF$ and $\triangle RST$:

• Calculate the ratios of corresponding sides. $\frac{DE}{RS}=\frac{16}{32}=\frac{1}{2}$, $\frac{EF}{ST}=\frac{34}{68}=\frac{1}{2}$, $\frac{DF}{RT}=\frac{30}{60}=\frac{1}{2}$.
• Since corresponding - side lengths are in proportion and we assume corresponding angles are congruent, the polygons are similar. The similarity statement is $\triangle DEF\sim\triangle RST$ and the scale factor is $\frac{1}{2}$.

Step6: Problem 5

For quadrilateral $PQRS$ and the other polygon:

• $\frac{PQ}{?}=\frac{5}{?}$, $\frac{QR}{GH}=\frac{9}{10}$, $\frac{RS}{HJ}=\frac{5}{8}$, $\frac{SP}{JK}=\frac{15}{25}=\frac{3}{5}$.
• Since the ratios of corresponding sides are not equal ($\frac{9}{10}

eq\frac{5}{8}
eq\frac{3}{5}$), the polygons are not similar.

Step7: Problem 6

For $\triangle ABC$ and $\triangle LMN$:

• Calculate the ratios of corresponding sides. $\frac{AB}{LM}=\frac{18}{6} = 3$, $\frac{BC}{MN}=\frac{12}{4}=3$, $\frac{CA}{NL}=\frac{18}{6}=3$.
• Since corresponding - side lengths are in proportion and we assume corresponding angles are congruent, the polygons are similar. The similarity statement is $\triangle ABC\sim\triangle LMN$ and the scale factor is $3$.

Answer:

1. Pairs of 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. Pairs of congruent angles: $\angle A\cong\angle H$, $\angle B\cong\angle J$, $\angle C\cong\angle K$, $\angle D\cong\angle L$; Extended proportion: $\frac{AB}{HJ}=\frac{BC}{JK}=\frac{CD}{KL}=\frac{DA}{LH}$
3. Similar, $WXYZ\sim ABCD$, scale factor $\frac{2}{3}$
4. Similar, $\triangle DEF\sim\triangle RST$, scale factor $\frac{1}{2}$
5. Not similar
6. Similar, $\triangle ABC\sim\triangle LMN$, scale factor $3$