Worksheet for determining triangle similarity using side lengths and angles.

To determine whether the triangles in each pair are similar, we need to use the criteria for triangle similarity: AA (Angle-Angle), SSS (Side-Side-Side), or SAS (Side-Angle-Side). Let's analyze each problem step by step.

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Problem 1

#### a)
- Triangles \( \triangle PQR \) and \( \triangle XYZ \)
- Side lengths:
- \( \triangle PQR \): \( PQ = 8 \), \( PR = 9 \), \( QR = 12 \)
- \( \triangle XYZ \): \( XY = 8 \), \( XZ = 6 \), \( YZ = 4 \)

Check the ratios of corresponding sides:
\[
\frac{PQ}{XY} = \frac{8}{8} = 1, \quad \frac{PR}{XZ} = \frac{9}{6} = 1.5, \quad \frac{QR}{YZ} = \frac{12}{4} = 3
\]
The ratios are not equal, so the triangles are not similar.

#### b)
- Triangles \( \triangle EFG \) and \( \triangle GHI \)
- Side lengths:
- \( \triangle EFG \): \( EF = 8 \), \( EG = 9 \), \( FG = 12 \)
- \( \triangle GHI \): \( GH = 18 \), \( GI = 16 \), \( HI = 24 \)

Check the ratios of corresponding sides:
\[
\frac{EF}{GH} = \frac{8}{18} = \frac{4}{9}, \quad \frac{EG}{GI} = \frac{9}{16}, \quad \frac{FG}{HI} = \frac{12}{24} = \frac{1}{2}
\]
The ratios are not equal, so the triangles are not similar.

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Problem 2

- Triangles \( \triangle ABC \) and \( \triangle QRP \)
- Angles:
- \( \triangle ABC \): \( \angle B = 90^\circ \), \( \angle A = 28^\circ \), \( \angle C = 62^\circ \)
- \( \triangle QRP \): \( \angle R = 90^\circ \), \( \angle Q = 62^\circ \), \( \angle P = 28^\circ \)

Both triangles have the same angles:
\[
\angle A = \angle P = 28^\circ, \quad \angle B = \angle R = 90^\circ, \quad \angle C = \angle Q = 62^\circ
\]
By the AA criterion, the triangles are similar.
\[
\triangle ABC \sim \triangle QRP
\]

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Problem 3

- Triangles \( \triangle ABC \) and \( \triangle DEC \)
- Angles and sides:
- \( \triangle ABC \): \( \angle A = 40^\circ \), \( \angle B = 60^\circ \), \( \angle C = 80^\circ \)
- \( \triangle DEC \): \( \angle D = 60^\circ \), \( \angle E = 40^\circ \), \( \angle C = 80^\circ \)

Both triangles share \( \angle C = 80^\circ \) and have the same other angles:
\[
\angle A = \angle E = 40^\circ, \quad \angle B = \angle D = 60^\circ
\]
By the AA criterion, the triangles are similar.
\[
\triangle ABC \sim \triangle DEC
\]

---

Problem 4

- Triangles \( \triangle LMN \) and \( \triangle PQR \)
- Side lengths:
- \( \triangle LMN \): \( LM = 30 \), \( MN = 32 \), \( LN = 24 \)
- \( \triangle PQR \): \( PQ = 48 \), \( QR = 45 \), \( PR = 36 \)

Check the ratios of corresponding sides:
\[
\frac{LM}{PQ} = \frac{30}{48} = \frac{5}{8}, \quad \frac{MN}{QR} = \frac{32}{45}, \quad \frac{LN}{PR} = \frac{24}{36} = \frac{2}{3}
\]
The ratios are not equal, so the triangles are not similar.

---

Problem 5

- Triangles \( \triangle ABC \) and \( \triangle XYZ \)
- Angles:
- \( \triangle ABC \): \( \angle B = 90^\circ \), \( \angle A = 30^\circ \), \( \angle C = 60^\circ \)
- \( \triangle XYZ \): \( \angle Y = 90^\circ \), \( \angle X = 30^\circ \), \( \angle Z = 60^\circ \)

Both triangles have the same angles:
\[
\angle A = \angle X = 30^\circ, \quad \angle B = \angle Y = 90^\circ, \quad \angle C = \angle Z = 60^\circ
\]
By the AA criterion, the triangles are similar.
\[
\triangle ABC \sim \triangle XYZ
\]

---

Problem 6

- Triangles \( \triangle KFL \) and \( \triangle MFG \)
- Angles:
- \( \triangle KFL \): \( \angle KFL = 54^\circ \), \( \angle FLK = 54^\circ \), \( \angle LKF = 72^\circ \)
- \( \triangle MFG \): \( \angle MFG = 54^\circ \), \( \angle FMG = 54^\circ \), \( \angle GMF = 72^\circ \)

Both triangles have the same angles:
\[
\angle KFL = \angle MFG = 54^\circ, \quad \angle FLK = \angle FMG = 54^\circ, \quad \angle LKF = \angle GMF = 72^\circ
\]
By the AA criterion, the triangles are similar.
\[
\triangle KFL \sim \triangle MFG
\]

---

Problem 7

- Triangles \( \triangle EFG \) and \( \triangle KLM \)
- Angles:
- \( \triangle EFG \): \( \angle E = 40^\circ \), \( \angle F = 100^\circ \), \( \angle G = 40^\circ \)
- \( \triangle KLM \): \( \angle K = 40^\circ \), \( \angle L = 100^\circ \), \( \angle M = 40^\circ \)

Both triangles have the same angles:
\[
\angle E = \angle K = 40^\circ, \quad \angle F = \angle L = 100^\circ, \quad \angle G = \angle M = 40^\circ
\]
By the AA criterion, the triangles are similar.
\[
\triangle EFG \sim \triangle KLM
\]

---

Final Answers

\[
\boxed{
\begin{array}{ll}
1a) & \text{Not similar} \\
1b) & \text{Not similar} \\
2) & \triangle ABC \sim \triangle QRP \\
3) & \triangle ABC \sim \triangle DEC \\
4) & \text{Not similar} \\
5) & \triangle ABC \sim \triangle XYZ \\
6) & \triangle KFL \sim \triangle MFG \\
7) & \triangle EFG \sim \triangle KLM \\
\end{array}
}
\]