Question

practice
example 1
list all pairs of congruent angles, and write a proportion that relates the corresponding sides for each pair of similar polygons.

1. (abcd sim wxyz)
2. (mnpq sim rstu)

example 2
determine whether each pair of figures is similar. if so, find the scale factor. explain your reasoning.
3.
4.
5.
6.

Explanation:

1. (ABCD ~ WXYZ)

Step1: Identify congruent angles

Corresponding angles of similar polygons are congruent:
$\angle A \cong \angle W$, $\angle B \cong \angle X$, $\angle C \cong \angle Y$, $\angle D \cong \angle Z$

Step2: Write side proportion

Corresponding sides are proportional:
$\frac{AB}{WX} = \frac{BC}{XY} = \frac{CD}{YZ} = \frac{DA}{ZW}$

2. (MNPQ ~ RSTU)

Step1: Identify congruent angles

Corresponding angles of similar polygons are congruent:
$\angle M \cong \angle R$, $\angle N \cong \angle S$, $\angle P \cong \angle T$, $\angle Q \cong \angle U$

Step2: Write side proportion

Corresponding sides are proportional:
$\frac{MN}{RS} = \frac{NP}{ST} = \frac{PQ}{TU} = \frac{QM}{UR}$

3. (Triangles WZY & MKL)

Step1: Verify angle congruence

For $\triangle WZY$: $\angle Z=90^\circ$, $\angle W=39^\circ$, $\angle Y=51^\circ$
For $\triangle MKL$: $\angle K=90^\circ$, $\angle M=42^\circ$, $\angle L=48^\circ$
No matching angle pairs, so angles are not congruent.

Step2: Conclusion on similarity

Since corresponding angles are not congruent, the triangles are not similar.

4. (Quadrilateral WXYZ)

Step1: Analyze angle/side relationships

$\triangle WXZ$ and $\triangle YXZ$ share side $XZ$, $WZ=YZ$, but no information about corresponding angles or proportional sides for the whole figure (no second polygon provided, assume comparing the two triangles):
$\angle WXZ
ot\cong \angle YXZ$ (no angle marks), and sides are not proportional in a way that satisfies similarity.

Step2: Conclusion on similarity

The two triangles (and thus no similar polygons here) are not similar, as there is no evidence of AA similarity or proportional sides with congruent included angles.

5. (Rectangles MNPL & GHJF)

Step1: Check angle congruence

All angles in rectangles are $90^\circ$, so all corresponding angles are congruent.

Step2: Calculate side ratios

Ratio of short sides: $\frac{MN}{GH} = \frac{2}{8} = \frac{1}{4}$
Ratio of long sides: $\frac{NP}{HJ} = \frac{4}{10} = \frac{2}{5}$

Step3: Conclusion on similarity

Since $\frac{1}{4}
eq \frac{2}{5}$, corresponding sides are not proportional. The rectangles are not similar.

6. (Triangles BDC & FGC)

Step1: Verify angle congruence

$\angle D = \angle G = 90^\circ$, $\angle BCD = \angle FCG$ (vertical angles are congruent). By AA similarity, the triangles are similar.

Step2: Calculate scale factor

Scale factor from $\triangle BDC$ to $\triangle FGC$:
$\frac{BD}{FG} = \frac{4}{\frac{20}{3}} = \frac{12}{20} = \frac{3}{5}$, or $\frac{DC}{CG} = \frac{3}{5}$, $\frac{BC}{FC} = \frac{5}{\frac{25}{3}} = \frac{15}{25} = \frac{3}{5}$

Answer:

1. Congruent angles: $\angle A \cong \angle W$, $\angle B \cong \angle X$, $\angle C \cong \angle Y$, $\angle D \cong \angle Z$; Proportion: $\frac{AB}{WX} = \frac{BC}{XY} = \frac{CD}{YZ} = \frac{DA}{ZW}$
2. Congruent angles: $\angle M \cong \angle R$, $\angle N \cong \angle S$, $\angle P \cong \angle T$, $\angle Q \cong \angle U$; Proportion: $\frac{MN}{RS} = \frac{NP}{ST} = \frac{PQ}{TU} = \frac{QM}{UR}$
3. Not similar; corresponding angles are not congruent.
4. Not similar; no evidence of proportional sides/congruent angles for similar polygons.
5. Not similar; side ratios are not equal.
6. Similar (AA similarity); scale factor from $\triangle BDC$ to $\triangle FGC$ is $\frac{3}{5}$