Question

practice
period date
example 1 practice • spiral review • extra practice
determine whether each pair of triangles is similar and explain your reasoning. if yes, then write a similarity statement.
1.
2.
3.
4.
ample 2
ntify the

Explanation:

Step1: Analyze Triangle Pair 1

Triangles $\triangle TRS$ and $\triangle XWS$:

1. $\angle TRS = \angle XWS$ (vertical angles, equal)
2. $\overline{TS} = \overline{XS}$, $\overline{RS} = \overline{WS}$ (marked congruent sides)

By SAS (Side-Angle-Side) Similarity (since included angle is equal and sides are in 1:1 ratio, so congruent, a subset of similar):
$\triangle TRS \sim \triangle XWS$

Step2: Analyze Triangle Pair 2

Triangles $\triangle ABC$ and $\triangle QPR$:
Calculate side ratios:
$\frac{AB}{QP} = \frac{8}{6} = \frac{4}{3}$
$\frac{BC}{PR} = \frac{12}{9} = \frac{4}{3}$
$\frac{AC}{QR} = \frac{12}{9} = \frac{4}{3}$
All corresponding sides are proportional (SSS Similarity):
$\triangle ABC \sim \triangle QPR$

Step3: Analyze Triangle Pair 3

Triangles $\triangle STU$ and $\triangle JPM$:
Calculate side ratios:
$\frac{ST}{JP} = \frac{10}{15} = \frac{2}{3}$
$\frac{SU}{JM} = \frac{14}{21} = \frac{2}{3}$
Included angles $\angle S = \angle J = 70^\circ$ (equal)
By SAS Similarity (included angle equal, sides proportional):
$\triangle STU \sim \triangle JPM$

Step4: Analyze Triangle Pair 4

Triangles $\triangle ABC$ and $\triangle EDF$:

1. $\overline{AB} = \overline{ED}$ (marked congruent)
2. $\overline{BC} = \overline{EF}$ (marked congruent)
3. $\angle A = \angle D = 90^\circ$ (right angles, equal)

By HL (Hypotenuse-Leg) for right triangles, they are congruent (so similar, 1:1 ratio):
$\triangle ABC \sim \triangle EDF$

Answer:

1. $\triangle TRS \sim \triangle XWS$ (SAS Similarity, vertical angle + congruent sides)
2. $\triangle ABC \sim \triangle QPR$ (SSS Similarity, all sides in $\frac{4}{3}$ ratio)
3. $\triangle STU \sim \triangle JPM$ (SAS Similarity, $70^\circ$ angle + sides in $\frac{2}{3}$ ratio)
4. $\triangle ABC \sim \triangle EDF$ (HL Congruence/Similarity, right triangles with congruent hypotenuse and leg)