Geometry - Similar Triangles - Guided Notes and Worksheet by Word ... - Free Printable

To solve the problem of determining whether the triangles in each pair are similar, we need to use the criteria for triangle similarity. The main criteria 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. 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.
3. SSS (Side-Side-Side): If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

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

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1.

- Triangles:
- Triangle on the left: Angles marked as \( \angle A \) and \( \angle B \).
- Triangle on the right: Angles marked as \( \angle D \) and \( \angle E \).

- Analysis:
- Both triangles have two pairs of corresponding angles marked as congruent (\( \angle A \cong \angle D \) and \( \angle B \cong \angle E \)).
- By the AA criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: AA

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2.

- Triangles:
- Triangle on the left: Sides marked as \( AB \), \( BC \), and \( AC \).
- Triangle on the right: Sides marked as \( DE \), \( EF \), and \( DF \).

- Analysis:
- The sides are not marked with any proportionality or angle congruence information.
- There is no clear indication of similarity based on the given markings.

- Answer:
- Similar: No
- Reason: Not enough information

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3.

- Triangles:
- Triangle on the left: Angles marked as \( \angle X \) and \( \angle Y \).
- Triangle on the right: Angles marked as \( \angle P \) and \( \angle Q \).

- Analysis:
- Both triangles have two pairs of corresponding angles marked as congruent (\( \angle X \cong \angle P \) and \( \angle Y \cong \angle Q \)).
- By the AA criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: AA

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4.

- Triangles:
- Triangle on the left: Sides marked as \( LM \), \( MN \), and \( LN \).
- Triangle on the right: Sides marked as \( PQ \), \( QR \), and \( PR \).

- Analysis:
- The sides are marked with proportionality symbols (\( LM \parallel PQ \), \( MN \parallel QR \), \( LN \parallel PR \)).
- This indicates that the sides are proportional.
- By the SSS criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: SSS

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5.

- Triangles:
- Triangle on the left: Angles marked as \( \angle R \) and \( \angle T \).
- Triangle on the right: Angles marked as \( \angle V \) and \( \angle W \).

- Analysis:
- Both triangles have two pairs of corresponding angles marked as congruent (\( \angle R \cong \angle V \) and \( \angle T \cong \angle W \)).
- By the AA criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: AA

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6.

- Triangles:
- Triangle on the left: Angles marked as \( \angle G \) and \( \angle H \).
- Triangle on the right: Angles marked as \( \angle J \) and \( \angle K \).

- Analysis:
- Both triangles have two pairs of corresponding angles marked as congruent (\( \angle G \cong \angle J \) and \( \angle H \cong \angle K \)).
- By the AA criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: AA

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7.

- Triangles:
- Triangle on the left: Sides marked as \( XY \), \( YZ \), and \( XZ \).
- Triangle on the right: Sides marked as \( UV \), \( VW \), and \( UW \).

- Analysis:
- The sides are not marked with any proportionality or angle congruence information.
- There is no clear indication of similarity based on the given markings.

- Answer:
- Similar: No
- Reason: Not enough information

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8.

- Triangles:
- Triangle on the left: Angles marked as \( \angle M \) and \( \angle N \).
- Triangle on the right: Angles marked as \( \angle P \) and \( \angle Q \).

- Analysis:
- Both triangles have two pairs of corresponding angles marked as congruent (\( \angle M \cong \angle P \) and \( \angle N \cong \angle Q \)).
- By the AA criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: AA

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9.

- Triangles:
- Triangle on the left: Sides marked as \( JK \), \( KL \), and \( JL \).
- Triangle on the right: Sides marked as \( OP \), \( PQ \), and \( OQ \).

- Analysis:
- The sides are not marked with any proportionality or angle congruence information.
- There is no clear indication of similarity based on the given markings.

- Answer:
- Similar: No
- Reason: Not enough information

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10.

- Triangles:
- Triangle on the left: Angles marked as \( \angle U \) and \( \angle V \).
- Triangle on the right: Angles marked as \( \angle X \) and \( \angle Y \).

- Analysis:
- Both triangles have two pairs of corresponding angles marked as congruent (\( \angle U \cong \angle X \) and \( \angle V \cong \angle Y \)).
- By the AA criterion, the triangles are similar.

- Answer:
- Similar: Yes
- Reason: AA

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Final Answers:

1. Similar: Yes, Reason: AA
2. Similar: No, Reason: Not enough information
3. Similar: Yes, Reason: AA
4. Similar: Yes, Reason: SSS
5. Similar: Yes, Reason: AA
6. Similar: Yes, Reason: AA
7. Similar: No, Reason: Not enough information
8. Similar: Yes, Reason: AA
9. Similar: No, Reason: Not enough information
10. Similar: Yes, Reason: AA

\[
\boxed{
\begin{array}{ll}
1. & \text{Yes, AA} \\
2. & \text{No, Not enough information} \\
3. & \text{Yes, AA} \\
4. & \text{Yes, SSS} \\
5. & \text{Yes, AA} \\
6. & \text{Yes, AA} \\
7. & \text{No, Not enough information} \\
8. & \text{Yes, AA} \\
9. & \text{No, Not enough information} \\
10. & \text{Yes, AA} \\
\end{array}
}
\]