Exercises: Triangle similarity criteria (AA, SAS, SSS) and proportional sides - ACT Math

A

Recall / Warm-Up

1.

What is the sum of the interior angles of any triangle?

A.

90°

B.

180°

C.

270°

D.

360°

2.

In a triangle, the angles measure 45° and 65°. What is the measure of the third angle?

A.

60°

B.

70°

C.

80°

D.

110°

3.

Solve for $x$ :
$\frac{3}{5} = \frac{x}{20}$

B

Fluency Practice

1.

In $\triangle ABC$ , $\angle A = 50\degree$ and $\angle B = 70\degree$ . In $\triangle DEF$ , $\angle D = 50\degree$ and $\angle E = 70\degree$ . Are the triangles similar? If so, by which criterion?

A.

Yes, by AA

B.

Yes, by SSS

C.

Yes, by SAS

D.

No, there is not enough information

2.

Triangle 1 has sides 4, 6, and 8. Triangle 2 has sides 6, 9, and 12. Are the triangles similar?

A.

Yes, by SSS — all three side ratios equal 2:3

B.

Yes, by SSS — all three side ratios equal 3:4

C.

No — only two side ratios are equal

D.

No — the triangles have different perimeters

3.

$\triangle ABC \sim \triangle DEF$ . If $AB = 10$ , $DE = 6$ , and $BC = 15$ , find $EF$ .

4.

$\triangle PQR \sim \triangle XYZ$ . If $PQ = 8$ , $QR = 12$ , $XY = 6$ , find $YZ$ .

5.

In $\triangle ABC$ , $AB = 12$ , $AC = 18$ , and $\angle A = 55\degree$ . In $\triangle DEF$ , $DE = 8$ , $DF = 12$ , and $\angle D = 55\degree$ . Are the triangles similar?

A.

Yes, by SAS — two pairs of sides are proportional and the included angles are equal

B.

Yes, by AA — the included angles are equal

C.

No — the side ratios are not equal

D.

No — SAS only works for congruence, not similarity

6.

$\triangle ABC \sim \triangle DEF$ with scale factor $k = 3$ (where $k = AB/DE$ ). If $AC = 21$ , find $DF$ .

C

Varied Practice

D

Word Problems / Application

1.

A 6-foot-tall person stands near a tree. The person casts a 4-foot shadow, and the tree casts a 30-foot shadow at the same time of day.

How tall is the tree, in feet?

2.

In $\triangle ABC$ , point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $\overline{DE} \parallel \overline{BC}$ . Given: $AD = 5$ , $DB = 10$ , and $AE = 4$ .

1.

What is the length of $EC$ ?

2.

What is the ratio of the area of $\triangle ADE$ to the area of $\triangle ABC$ ? Express your answer as a simplified fraction.

3.

Two similar triangular garden plots have perimeters of 24 m and 36 m. The smaller plot has an area of 32 m².

What is the area of the larger plot, in square meters?

E

Error Analysis

1.

Kim solved this problem:

" $\triangle ABC \sim \triangle DEF$ . $AB = 8$ , $BC = 12$ , $DE = 6$ . Find $DF$ ."

Kim's work:

$\frac{AB}{DE} = \frac{BC}{DF}$
$\frac{8}{6} = \frac{12}{DF}$
$8 \times DF = 72$
$DF = 9$

What error did Kim make, and what is the correct answer?

A.

Kim matched $BC$ with $DF$ , but corresponding sides are $BC \leftrightarrow EF$ and $AC \leftrightarrow DF$ . You cannot find $DF$ without knowing $AC$ .

B.

Kim cross-multiplied incorrectly. The correct answer is 10.

C.

Kim's work is completely correct. DF = 9.

D.

Kim set up the ratios upside down. The correct answer is 16.

2.

Marcus solved this problem:

"Two similar triangles have sides in ratio 3:1. The smaller triangle has area 10 cm². Find the area of the larger triangle."

Marcus's work:

Scale factor = 3
Larger area = 10 × 3 = 30 cm²

What error did Marcus make, and what is the correct area?

A.

Marcus multiplied area by the scale factor instead of the square of the scale factor. The correct area is 90 cm².

B.

Marcus should have divided by 3 instead of multiplying. The correct area is 10/3 cm².

C.

Marcus should have used $k^3 = 27$ . The correct area is 270 cm².

D.

Marcus's answer of 30 cm² is correct.

F

Challenge / Extension

1.

In $\triangle ABC$ , $\overline{DE} \parallel \overline{BC}$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ . The area of $\triangle ADE$ is 16 cm² and the area of trapezoid $BCED$ is 48 cm².

What is the ratio $AD:DB$ ? Express your answer as a simplified fraction.

2.