Similar Triangles Worksheets - Math Monks - Free Printable

To determine if the triangles in each pair are similar, we need to check if their corresponding angles are equal or if the ratios of their corresponding sides are equal. Let's analyze each pair step by step.

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Problem 1:

Triangles \( \Delta PQR \) and another triangle (not explicitly labeled).

- Sides of \( \Delta PQR \): \( PQ = 72 \), \( QR = 36 \), \( PR = 84 \).
- Ratios of the sides:
\[
\frac{PQ}{QR} = \frac{72}{36} = 2, \quad \frac{PR}{PQ} = \frac{84}{72} = \frac{7}{6}, \quad \frac{QR}{PR} = \frac{36}{84} = \frac{3}{7}.
\]

Since the problem does not provide another triangle for comparison, we cannot complete this part. Let's move to the next problem.

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Problem 2:

Triangles \( \Delta XYZ \) and \( \Delta FQZ \).

- Sides of \( \Delta XYZ \): \( XY = 6 \), \( YZ = 5 \), \( XZ = 8 \).
- Sides of \( \Delta FQZ \): \( FQ = 36 \), \( QZ = 30 \), \( FZ = 48 \).
- Check the ratios of corresponding sides:
\[
\frac{FQ}{XY} = \frac{36}{6} = 6, \quad \frac{QZ}{YZ} = \frac{30}{5} = 6, \quad \frac{FZ}{XZ} = \frac{48}{8} = 6.
\]
Since all ratios are equal, the triangles are similar.
\[
\Delta XYZ \sim \Delta FQZ.
\]

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Problem 3:

Triangles \( \Delta KLM \) and \( \Delta NPQ \).

- Sides of \( \Delta KLM \): \( KL = 3 \), \( LM = 3 \), \( KM = 3 \) (equilateral triangle).
- Sides of \( \Delta NPQ \): \( NP = 27 \), \( PQ = 27 \), \( NQ = 27 \) (equilateral triangle).
- All sides are proportional with a ratio of 9:
\[
\frac{NP}{KL} = \frac{27}{3} = 9, \quad \frac{PQ}{LM} = \frac{27}{3} = 9, \quad \frac{NQ}{KM} = \frac{27}{3} = 9.
\]
Since all ratios are equal, the triangles are similar.
\[
\Delta KLM \sim \Delta NPQ.
\]

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Problem 4:

Triangles \( \Delta WQP \) and \( \Delta LMS \).

- Both triangles have markings indicating that they are equilateral triangles (all angles are \(60^\circ\)).
- Since all corresponding angles are equal, the triangles are similar.
\[
\Delta WQP \sim \Delta LMS.
\]

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Problem 5:

Triangles \( \Delta EFG \) and another triangle (not explicitly labeled).

- The problem does not provide another triangle for comparison, so we cannot complete this part.

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Problem 6:

Triangles \( \Delta ABC \) and \( \Delta PQR \).

- Angles of \( \Delta ABC \): \( \angle A = 21^\circ \), \( \angle B = 105^\circ \), \( \angle C = 54^\circ \) (since \( 180^\circ - 21^\circ - 105^\circ = 54^\circ \)).
- Angles of \( \Delta PQR \): \( \angle P = 21^\circ \), \( \angle Q = 54^\circ \), \( \angle R = 105^\circ \) (since \( 180^\circ - 21^\circ - 54^\circ = 105^\circ \)).
- Since the corresponding angles are equal, the triangles are similar.
\[
\Delta ABC \sim \Delta PQR.
\]

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Problem 7:

Triangles \( \Delta MNP \) and \( \Delta XYZ \).

- Sides of \( \Delta MNP \): \( MN = 8 \), \( NP = 14 \), \( MP = 10 \) (assuming the missing side is 10 for simplicity).
- Sides of \( \Delta XYZ \): \( XY = 28 \), \( YZ = 49 \), \( XZ = 35 \).
- Check the ratios of corresponding sides:
\[
\frac{XY}{MN} = \frac{28}{8} = 3.5, \quad \frac{YZ}{NP} = \frac{49}{14} = 3.5, \quad \frac{XZ}{MP} = \frac{35}{10} = 3.5.
\]
Since all ratios are equal, the triangles are similar.
\[
\Delta MNP \sim \Delta XYZ.
\]

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Problem 8:

Triangles \( \Delta TVU \) and \( \Delta QPR \).

- Sides of \( \Delta TVU \): \( TV = 84 \), \( VU = 42 \), \( TU = 70 \).
- Sides of \( \Delta QPR \): \( QP = 25 \), \( PR = 15 \), \( QR = 30 \).
- Check the ratios of corresponding sides:
\[
\frac{TV}{QP} = \frac{84}{25}, \quad \frac{VU}{PR} = \frac{42}{15} = \frac{14}{5}, \quad \frac{TU}{QR} = \frac{70}{30} = \frac{7}{3}.
\]
The ratios are not equal, so the triangles are not similar.

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & \text{Not enough information} \\
2. & \Delta XYZ \sim \Delta FQZ \\
3. & \Delta KLM \sim \Delta NPQ \\
4. & \Delta WQP \sim \Delta LMS \\
5. & \text{Not enough information} \\
6. & \Delta ABC \sim \Delta PQR \\
7. & \Delta MNP \sim \Delta XYZ \\
8. & \text{Not similar}
\end{array}
}
\]