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 \( \Delta FQZ \): \( FQ = 36 \), \( QZ = 30 \), \( FZ = 48 \).
- Ratios of the 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 corresponding side 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).
- Ratios of the sides:
\[
\frac{NP}{KL} = \frac{27}{3} = 9, \quad \frac{PQ}{LM} = \frac{27}{3} = 9, \quad \frac{NQ}{KM} = \frac{27}{3} = 9.
\]
Since all corresponding side 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 equiangular (all angles are equal).
- Since the triangles are equiangular, they 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 \( \Delta ABC \): \( \angle A = 21^\circ \), \( \angle B = 105^\circ \), \( \angle C = 180^\circ - 21^\circ - 105^\circ = 54^\circ \).
- Angles of \( \Delta PQR \): \( \angle P = 21^\circ \), \( \angle Q = 54^\circ \), \( \angle R = 180^\circ - 21^\circ - 54^\circ = 105^\circ \).
- 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 \( \Delta MNP \): \( MN = 8 \), \( NP = 14 \), \( MP = \text{(not given)} \).
- Sides of \( \Delta XYZ \): \( XY = 28 \), \( YZ = 28 \), \( XZ = 49 \).
- Ratios of the sides:
\[
\frac{XY}{MN} = \frac{28}{8} = \frac{7}{2}, \quad \frac{YZ}{NP} = \frac{28}{14} = 2, \quad \frac{XZ}{MP} = \text{(cannot be determined without \( MP \))}.
\]
Since the ratios are not equal, the triangles are not similar.

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Problem 8:

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

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

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Final Answers:

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