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

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

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Problem 1:

- Triangles: ΔABC and ΔDEF
- Given: ∠A ≅ ∠D, ∠B ≅ ∠E

Since two pairs of corresponding angles are congruent, we can conclude that the triangles are similar by the AA criterion.

Similarity Statement:
ΔABC ~ ΔDEF (AA)

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Problem 2:

- Triangles: ΔPQR and ΔXYZ
- Given: PQ/XY = QR/YZ = PR/XZ

Since the ratios of the corresponding sides are equal, we can conclude that the triangles are similar by the SSS criterion.

Similarity Statement:
ΔPQR ~ ΔXYZ (SSS)

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Problem 3:

- Triangles: ΔMNO and ΔSTU
- Given: MN/ST = MO/SU and ∠M ≅ ∠S

Since the ratios of two pairs of corresponding sides are equal and the included angles are congruent, we can conclude that the triangles are similar by the SAS criterion.

Similarity Statement:
ΔMNO ~ ΔSTU (SAS)

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Problem 4:

- Triangles: ΔJKL and ΔXYZ
- Given: ∠J ≅ ∠X, ∠K ≅ ∠Y

Since two pairs of corresponding angles are congruent, we can conclude that the triangles are similar by the AA criterion.

Similarity Statement:
ΔJKL ~ ΔXYZ (AA)

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Problem 5:

- Triangles: ΔGHI and ΔLMN
- Given: GH/LM = HI/MN = GI/LN

Since the ratios of the corresponding sides are equal, we can conclude that the triangles are similar by the SSS criterion.

Similarity Statement:
ΔGHI ~ ΔLMN (SSS)

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Problem 6:

- Triangles: ΔRST and ΔUVW
- Given: RS/UV = RT/UW and ∠R ≅ ∠U

Since the ratios of two pairs of corresponding sides are equal and the included angles are congruent, we can conclude that the triangles are similar by the SAS criterion.

Similarity Statement:
ΔRST ~ ΔUVW (SAS)

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Final Answer:

\[
\boxed{
\begin{array}{l}
\text{1. } \Delta ABC \sim \Delta DEF \text{ (AA)} \\
\text{2. } \Delta PQR \sim \Delta XYZ \text{ (SSS)} \\
\text{3. } \Delta MNO \sim \Delta STU \text{ (SAS)} \\
\text{4. } \Delta JKL \sim \Delta XYZ \text{ (AA)} \\
\text{5. } \Delta GHI \sim \Delta LMN \text{ (SSS)} \\
\text{6. } \Delta RST \sim \Delta UVW \text{ (SAS)}
\end{array}
}
\]