basic proof worksheet.pdf - Geometry Proof Worksheet #2 Name - Free Printable

Problem Overview:

The task involves completing proofs for four different scenarios where triangles are proven to be congruent using geometric postulates and theorems. Each proof requires filling in missing statements and reasons based on given information.

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Proof 1: Prove ΔXYZ ≅ ΔXST

#### Given:
- ∠Y ≅ ∠S
- XY ≅ XS
- XZ ≅ XT

#### Goal:
Prove that ΔXYZ ≅ ΔXST.

#### Proof Steps:

1. Statement: XY ≅ XS
Reason: Given

2. Statement: ∠Y ≅ ∠S
Reason: Given

3. Statement: XZ ≅ XT
Reason: Given

4. Statement: ΔXYZ ≅ ΔXST
Reason: Side-Angle-Side (SAS) Congruence Postulate

#### Explanation:
- The SAS Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
- Here, we have:
- XY ≅ XS (side)
- ∠Y ≅ ∠S (angle between the sides)
- XZ ≅ XT (side)
- Therefore, by SAS, ΔXYZ ≅ ΔXST.

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Proof 2: Prove ΔABC ≅ ΔECD

#### Given:
- AB || ED
- AC || EF
- BC || DF

#### Goal:
Prove that ΔABC ≅ ΔECD.

#### Proof Steps:

1. Statement: AB || ED
Reason: Given

2. Statement: AC || EF
Reason: Given

3. Statement: BC || DF
Reason: Given

4. Statement: ∠A ≅ ∠E
Reason: Alternate Interior Angles Theorem (since AB || ED)

5. Statement: ∠C ≅ ∠D
Reason: Alternate Interior Angles Theorem (since AC || EF)

6. Statement: ∠B ≅ ∠F
Reason: Alternate Interior Angles Theorem (since BC || DF)

7. Statement: ΔABC ≅ ΔECD
Reason: Angle-Angle-Angle (AAA) Similarity Postulate (Note: This is not a congruence postulate, but if additional side information were provided, it could lead to congruence.)

#### Explanation:
- The problem as stated does not provide enough information to prove congruence directly using standard congruence postulates (SSS, SAS, ASA, AAS). However, the given information shows that corresponding angles are congruent due to parallel lines.
- If additional side information were provided (e.g., corresponding sides are congruent), we could use a congruence postulate like ASA or AAS. Without such information, we can only conclude similarity, not congruence.

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Proof 3: Prove ΔLMN ≅ ΔOPQ

#### Given:
- LM ≅ OP
- LN ≅ OQ
- MN ≅ PQ

#### Goal:
Prove that ΔLMN ≅ ΔOPQ.

#### Proof Steps:

1. Statement: LM ≅ OP
Reason: Given

2. Statement: LN ≅ OQ
Reason: Given

3. Statement: MN ≅ PQ
Reason: Given

4. Statement: ΔLMN ≅ ΔOPQ
Reason: Side-Side-Side (SSS) Congruence Postulate

#### Explanation:
- The SSS Congruence Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
- Here, we have:
- LM ≅ OP
- LN ≅ OQ
- MN ≅ PQ
- Therefore, by SSS, ΔLMN ≅ ΔOPQ.

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Proof 4: Prove ΔLMN ≅ ΔLNP

#### Given:
- LM ≅ LP
- MN ≅ NP
- ∠MLN ≅ ∠PLN

#### Goal:
Prove that ΔLMN ≅ ΔLNP.

#### Proof Steps:

1. Statement: LM ≅ LP
Reason: Given

2. Statement: MN ≅ NP
Reason: Given

3. Statement: ∠MLN ≅ ∠PLN
Reason: Given

4. Statement: ΔLMN ≅ ΔLNP
Reason: Side-Angle-Side (SAS) Congruence Postulate

#### Explanation:
- The SAS Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
- Here, we have:
- LM ≅ LP (side)
- ∠MLN ≅ ∠PLN (angle between the sides)
- MN ≅ NP (side)
- Therefore, by SAS, ΔLMN ≅ ΔLNP.

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Final Answer:

\[
\boxed{
\text{Proof 1: SAS, Proof 2: AAA (Similarity), Proof 3: SSS, Proof 4: SAS}
}
\]