Question

similar polygons
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:

1.

Step1: Identify congruent angles for \(\triangle XYZ\sim\triangle MNP\)

By the definition of similar - triangles, \(\angle X\cong\angle M\), \(\angle Y\cong\angle N\), \(\angle Z\cong\angle P\)

Step2: Write the extended proportion for corresponding sides

\(\frac{XY}{MN}=\frac{YZ}{NP}=\frac{XZ}{MP}\)
2.

Step1: Identify congruent angles for \(ABCD\sim HJKL\)

\(\angle A\cong\angle H\), \(\angle B\cong\angle J\), \(\angle C\cong\angle K\), \(\angle D\cong\angle L\)

Step2: Write the extended proportion for corresponding sides

\(\frac{AB}{HJ}=\frac{BC}{JK}=\frac{CD}{KL}=\frac{DA}{LH}\)
3.

Step1: Check the ratio of corresponding sides for \(WXYZ\) and \(ABCD\)

\(\frac{WX}{AB}=\frac{10}{15}=\frac{2}{3}\), \(\frac{XY}{BC}=\frac{4}{6}=\frac{2}{3}\), \(\frac{YZ}{CD}=\frac{10}{15}=\frac{2}{3}\), \(\frac{ZW}{DA}=\frac{4}{6}=\frac{2}{3}\)

Step2: Check the congruence of corresponding angles

Since the quadrilaterals are parallelograms, corresponding angles are congruent.

Step3: Write the similarity statement and scale - factor

The similarity statement is \(WXYZ\sim ABCD\) and the scale factor is \(\frac{2}{3}\)
4.

Step1: Check the ratio of corresponding sides for \(\triangle DEF\) and \(\triangle RST\)

\(\frac{DE}{RS}=\frac{16}{32}=\frac{1}{2}\), \(\frac{EF}{ST}=\frac{34}{68}=\frac{1}{2}\), \(\frac{DF}{RT}=\frac{30}{60}=\frac{1}{2}\)

Step2: Check the congruence of corresponding angles

Since the right - angled triangles have congruent non - right angles (by AA similarity as the right angles are equal and the other non - right angles are equal because of the equal ratios of sides), the triangles are similar.

Step3: Write the similarity statement and scale - factor

The similarity statement is \(\triangle DEF\sim\triangle RST\) and the scale factor is \(\frac{1}{2}\)
5.

Step1: Check the ratio of corresponding sides for \(PQRS\) and \(GHKJ\)

\(\frac{PQ}{GH}\), \(\frac{QR}{HK}\), \(\frac{RS}{KJ}\), \(\frac{SP}{JG}\)
\(\frac{PQ}{GH}\) is not equal to \(\frac{QR}{HK}\) (for example, \(\frac{5}{10}
eq\frac{9}{25}\))
So the polygons are not similar.
6.

Step1: Check the ratio of corresponding sides for \(\triangle ABC\) and \(\triangle LMN\)

\(\frac{AB}{LM}=\frac{18}{6} = 3\), \(\frac{BC}{MN}=\frac{12}{4}=3\), \(\frac{CA}{NL}=\frac{18}{6}=3\)

Step2: Check the congruence of corresponding angles

Since the angles are congruent (by AA similarity as the ratios of sides are equal), the triangles are similar.

Step3: Write the similarity statement and scale - factor

The similarity statement is \(\triangle ABC\sim\triangle LMN\) and the scale factor is \(3\)

Answer:

1. Congruent angles: \(\angle X\cong\angle M\), \(\angle Y\cong\angle N\), \(\angle Z\cong\angle P\); Proportion: \(\frac{XY}{MN}=\frac{YZ}{NP}=\frac{XZ}{MP}\)
2. Congruent angles: \(\angle A\cong\angle H\), \(\angle B\cong\angle J\), \(\angle C\cong\angle K\), \(\angle D\cong\angle L\); 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 \(3\)