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# WS: Two-Column Proofs - geometry proofs practice with answers

Proving Triangles are Congruent

For each of the following, write a two-column proof.
|1. |2. |
|Given: [pic] |Given: [pic] |
|[pic] |[pic] |
|[pic] |Prove: [pic] |
|Prove: [pic] | |
| | |
|3. |4. |
|Given: [pic] bisects [pic] |Given: [pic] |
|[pic] |[pic] |
|Prove: ∆ATL ( ∆ATS |[pic] |
| |Prove:[pic] |
|5. |6. |
|Given: (B ( (C |Given: X is the midpoint of [pic] |
|(A ( (O |(A ( (I |
|[pic] |[pic] |
|Prove: [pic] |Prove: [pic] |
| | |
| | |
|7. |8. |
|Given: [pic] |Given: [pic] |
|[pic] |[pic] |
|Prove: [pic] |Prove: ∆PAT ( ∆PET |
| | |
| | |
|9. |10. |
|Given: [pic]bisects (LAN |Given: (KWL ( (ALW |
|[pic] |(AWL ( (WLK |
|Prove: [pic] |Prove: ∆KWL ( ∆ALW |
| | |

Proofs with CPCTC

For each of the following, write a two-column proof.

|1. |2. |
|Given: [pic] |Given: [pic] |
|[pic] |[pic] |
|Prove: [pic] |Prove: [pic] |

|3. |4. |
|Given: [pic] bisects [pic] |Given: I is the midpoint of [pic] |
|[pic] bisects [pic] |I is the midpoint of [pic] |
|Prove: [pic] |Prove: [pic] |
| | |
|5. |6. |
|Given: [pic] |Given: [pic] |
|[pic] bisects [pic] |[pic] |
|Prove: [pic] |Prove: [pic] |
| | |
|7. |8. |
|Given: [pic] |Given: [pic] |
|[pic] |[pic] |
|Prove: R is the midpoint of [pic] |Prove: ∆TOP ( ∆ZAP |
| | |
|9. |10. |
|Given: [pic] |Given: [pic] |
|[pic] |[pic] |
|Prove: [pic] |Prove: [pic] |
| | |

Proving Triangle Theorems

Complete a two-column proof for each of the following theorems.

| | |
|Third Angle Theorem: If two angles in one triangle are equal in measure to two |Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal |
|angles of another triangle, then the third angle in each triangle is equal in |to the sum of the two remote interior angles. |
|measure to the third angle in the other triangle. | |
| | |
| | |
|Given: [pic] | |
|[pic] | |
|Prove: [pic] | |
| |Prove: [pic] |
| | |
|Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the |Isosceles Triangle Converse: If two angles of a triangle are congruent, then the |
|angles opposite those sides are congruent. |triangle is isosceles. |
| | |
| | |
|Given: [pic] |Given: [pic] |
|[pic] is the median of [pic] |[pic] bisects [pic] |
|Prove: [pic] |Prove: [pic]is isosceles |
| | |
|Corollary: If a triangle is equilateral, then the angles are congruent. |Corollary: If the three angles of a triangle are congruent, then the triangle is |
| |equilateral. |
| | |
|Given: [pic] is equilateral | |
|Prove: [pic] |Given: [pic] |
| |Prove: [pic] is equilateral. |

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How do you write a proof in geometry? Geometric proofs can be written in one of two ways: two columns, or a paragraph. A paragraph proof is only a two-column proof written in sentences. However, since it is easier to leave steps out when writing a paragraph proof, we'll learn the two-column method.