**Home** / **geometry proofs practice with answers** / WS: Two-Column Proofs

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.