Similar Triangles Worksheets - Free Printable

To solve the problem of completing the similarity statements for the given triangles, we need to use the properties of similar triangles. Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. The common criteria for triangle similarity are:

1. AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
2. SAS (Side-Angle-Side): If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, the triangles are similar.
3. SSS (Side-Side-Side): If all three sides of one triangle are proportional to all three sides of another triangle, the triangles are similar.

Let's analyze each pair of triangles step by step:

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Problem 1:

[Image shows two triangles with marked angles.]

- Triangle on the left: Angles are marked as \( \angle A = 40^\circ \), \( \angle B = 70^\circ \).
- Triangle on the right: Angles are marked as \( \angle D = 40^\circ \), \( \angle E = 70^\circ \).

Since both triangles have two pairs of corresponding angles equal (\( \angle A = \angle D \) and \( \angle B = \angle E \)), the third angles must also be equal by the Angle Sum Property of triangles. Therefore, the triangles are similar by the AA criterion.

Similarity Statement:
\[
\triangle ABC \sim \triangle DEF \quad \text{(AA)}
\]

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Problem 2:

[Image shows two triangles with marked sides and an included angle.]

- Triangle on the left: Sides are marked as \( AB = 6 \), \( BC = 8 \), and the included angle \( \angle B = 50^\circ \).
- Triangle on the right: Sides are marked as \( DE = 9 \), \( EF = 12 \), and the included angle \( \angle E = 50^\circ \).

We observe that:
\[
\frac{AB}{DE} = \frac{6}{9} = \frac{2}{3}, \quad \frac{BC}{EF} = \frac{8}{12} = \frac{2}{3}
\]
The ratio of the corresponding sides is the same, and the included angles are equal (\( \angle B = \angle E \)). Therefore, the triangles are similar by the SAS criterion.

Similarity Statement:
\[
\triangle ABC \sim \triangle DEF \quad \text{(SAS)}
\]

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Problem 3:

[Image shows two triangles with all sides marked.]

- Triangle on the left: Sides are marked as \( AB = 5 \), \( BC = 10 \), \( AC = 15 \).
- Triangle on the right: Sides are marked as \( DE = 10 \), \( EF = 20 \), \( DF = 30 \).

We observe that:
\[
\frac{AB}{DE} = \frac{5}{10} = \frac{1}{2}, \quad \frac{BC}{EF} = \frac{10}{20} = \frac{1}{2}, \quad \frac{AC}{DF} = \frac{15}{30} = \frac{1}{2}
\]
All corresponding sides are in the same ratio. Therefore, the triangles are similar by the SSS criterion.

Similarity Statement:
\[
\triangle ABC \sim \triangle DEF \quad \text{(SSS)}
\]

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Problem 4:

[Image shows two triangles with marked angles.]

- Triangle on the left: Angles are marked as \( \angle A = 30^\circ \), \( \angle C = 80^\circ \).
- Triangle on the right: Angles are marked as \( \angle D = 30^\circ \), \( \angle F = 80^\circ \).

Since both triangles have two pairs of corresponding angles equal (\( \angle A = \angle D \) and \( \angle C = \angle F \)), the third angles must also be equal by the Angle Sum Property of triangles. Therefore, the triangles are similar by the AA criterion.

Similarity Statement:
\[
\triangle ABC \sim \triangle DEF \quad \text{(AA)}
\]

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Final Answer:

\[
\boxed{\triangle ABC \sim \triangle DEF \quad \text{(AA)}, \quad \triangle ABC \sim \triangle DEF \quad \text{(SAS)}, \quad \triangle ABC \sim \triangle DEF \quad \text{(SSS)}, \quad \triangle ABC \sim \triangle DEF \quad \text{(AA)}}
\]