Similar Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Problem 1: Prove that \( \triangle GHI \sim \triangle JKL \)

#### Given:
- \( \angle G \cong \angle J \)
- \( \frac{GH}{JK} = \frac{GI}{JL} \)

#### To Prove:
- \( \triangle GHI \sim \triangle JKL \)

#### Proof:

| Statements | Reasons |
|------------|---------|
| 1. \( \angle G \cong \angle J \) | 1. Given |
| 2. \( \frac{GH}{JK} = \frac{GI}{JL} \) | 2. Given |
| 3. \( \triangle GHI \sim \triangle JKL \) | 3. SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the sides including these angles are proportional, then the triangles are similar. |

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Problem 2: Prove that \( \triangle PRQ \sim \triangle SRT \)

#### Given:
- \( \overline{PQ} \parallel \overline{ST} \)

#### To Prove:
- \( \triangle PRQ \sim \triangle SRT \)

#### Proof:

| Statements | Reasons |
|------------|---------|
| 1. \( \overline{PQ} \parallel \overline{ST} \) | 1. Given |
| 2. \( \angle RPQ \cong \angle RTS \) | 2. Corresponding Angles Postulate: When two parallel lines are cut by a transversal, corresponding angles are congruent. |
| 3. \( \angle PRQ \cong \angle SRT \) | 3. Vertical Angles Theorem: Vertical angles are always congruent. |
| 4. \( \triangle PRQ \sim \triangle SRT \) | 4. AA Similarity Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. |

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Problem 3: Prove that \( \triangle ABC \sim \triangle AED \)

#### Given:
- \( \frac{AB}{AE} = \frac{AC}{AD} \)

#### To Prove:
- \( \triangle ABC \sim \triangle AED \)

#### Proof:

| Statements | Reasons |
|------------|---------|
| 1. \( \frac{AB}{AE} = \frac{AC}{AD} \) | 1. Given |
| 2. \( \angle BAC \cong \angle EAD \) | 2. Reflexive Property of Congruence: The same angle is congruent to itself. |
| 3. \( \triangle ABC \sim \triangle AED \) | 3. SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the sides including these angles are proportional, then the triangles are similar. |

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Final Answers:

1. \( \boxed{\text{SAS Similarity Theorem}} \)
2. \( \boxed{\text{AA Similarity Theorem}} \)
3. \( \boxed{\text{SAS Similarity Theorem}} \)