Complete the two-column proof. Given: ∠2 and ∠5 are supplementary Prove: Line l is above line m. Angle 1 is above line l and to the left of the transversal. Angle 2 is above line l and to the right of the transversal. Angle 3 is below line l and to the left of the transversal. Angle 4 is below line l and to the right of the transversal. Angle 5 is above line m and to the left of the transversal. Angle 6 is above line m and to the right of the transversal. Angle 7 is below line m and to the left of the transversal. Angle 8 is below line m and to the right of the transversal. Statements Reasons 1. _____________ 1. _____________ 2. ∠ 3 ∠ 2 2. _____________ 3. ∠ 3 and ∠ 5 are supplementary 3. _____________ 4. _____________ 4. _____________(6 points)
Statements Reasons
1. Angle 1 is above line l and to the left of the transversal. Given
2. Angle 2 is above line l and to the right of the transversal. Given
3. Angle 3 is below line l and to the left of the transversal. Given
4. Angle 4 is below line l and to the right of the transversal. Given
5. Angle 5 is above line m and to the left of the transversal. Given
6. Angle 6 is above line m and to the right of the transversal. Given
7. Angle 7 is below line m and to the left of the transversal. Given
8. Angle 8 is below line m and to the right of the transversal. Given
9. ∠2 and ∠5 are supplementary. Given
10. Angle 2 and angle 5 are on the same line l. Definition of supplementary angles
11. Line l is above line m. If two angles and their supplements form a straight line, then the lines containing those angles are parallel (Alternatively, this can be proven by contradiction using the angles on the same line theorem).
12. Line l is above line m. Conclusion (line 11).
Statements Reasons
1. Angle 1 is above line l and to the left of the transversal Given
2. Angle 2 is above line l and to the right of the transversal Given
3. Angle 3 is below line l and to the left of the transversal Given
4. Angle 4 is below line l and to the right of the transversal Given
5. Angle 5 is above line m and to the left of the transversal Given
6. Angle 6 is above line m and to the right of the transversal Given
7. Angle 7 is below line m and to the left of the transversal Given
Angle 2 and Angle 5 are supplementary Definition of supplementary angles
∠2 and ∠5 are supplementary Given
Angle 3 and Angle 5 are supplementary Definition of supplementary angles
∠3 and ∠5 are supplementary Given
Angle 4 and Angle 2 are supplementary Definition of supplementary angles
∠4 and ∠2 are supplementary Given
Angle 7 and Angle 5 are supplementary Definition of supplementary angles
∠7 and ∠5 are supplementary Given
Angle 1 and Angle 3 are supplementary Definition of supplementary angles
∠1 and ∠3 are supplementary Given
Angle 6 and Angle 4 are supplementary Definition of supplementary angles
∠6 and ∠4 are supplementary Given
Angle 8 and Angle 7 are supplementary Definition of supplementary angles
∠8 and ∠7 are supplementary Given
Angle 8 and Angle 6 are supplementary Definition of supplementary angles
∠8 and ∠6 are supplementary Given
Angle 1 and Angle 2 are supplementary Given
∠1 and ∠2 are supplementary Given
Line l is above line m By the Vertical Angle Theorem
To complete the two-column proof, we need to provide statements and reasons for each step.
Here is the completed proof:
Statements Reasons
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1. Angle 2 is above line l Given
2. Angle 5 is above line m Given
3. ∠2 and ∠5 are supplementary Given
4. Angle 1 is above line l and to Given, and definition of supplementary angles
the left of the transversal
5. Angle 3 is below line l and to Given, and definition of supplementary angles
the left of the transversal
6. Angle 7 is below line m and to Given, and definition of supplementary angles
the left of the transversal
7. Angle 8 is below line m and to Given, and definition of supplementary angles
the right of the transversal
8. Angle 6 is above line m and to Given, and definition of supplementary angles
the right of the transversal
9. Angle 4 is below line l and to Given, and definition of supplementary angles
the right of the transversal
10. Line l is above line m All angles being on one side of the transversal
and angles 2 and 5 being supplementary imply
that line l is above line m.
Note: This is just one possible approach to complete the proof. There might be alternative ways to prove the same statement, but this is a logical and valid proof based on the given information.