Write a paragraph proof of Theorem 3-8: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
The information and diagram for a two column proof is shown.
Given: Line AB is perpendicular to line CD
Line AE is perpendicular to line CD
Prove: Line AB is parallel to line AE
Proof:
Statements Reasons
1. Line AB is perpendicular to line CD Given
2. Line AE is perpendicular to line CD Given
3. ∠ABD and ∠AED are right angles Definition of perpendicular lines
4. ∠ABD ≅ ∠AED Definition of congruent angles
5. AB || AE Definition of parallel lines
In step 1, we are given that line AB is perpendicular to line CD, and in step 2, we are given that line AE is perpendicular to line CD. In step 3, we use the definition of perpendicular lines to state that ∠ABD and ∠AED are right angles. In step 4, we use the definition of congruent angles to state that ∠ABD and ∠AED are congruent. Finally, in step 5, we use the definition of parallel lines to conclude that line AB is parallel to line AE. Therefore, the theorem is proven.
Given:
Line AB is perpendicular to line CD
Line EF is perpendicular to line CD
To prove:
Line AB is parallel to line EF
Proof:
1. Draw a diagram with line AB and line EF intersecting line CD at point P and point Q, respectively.
2. Assume towards contradiction that line AB is not parallel to line EF.
3. Since line AB is perpendicular to line CD, angle APB is a right angle (90 degrees).
4. Likewise, since line EF is perpendicular to line CD, angle EQF is also a right angle (90 degrees).
5. Suppose lines AB and EF are not parallel. Then, angle APB and angle EQF will not be congruent.
6. By the Converse of the Corresponding Angles' Postulate, if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
7. However, this contradicts our assumption that line AB and line EF are not parallel.
8. Therefore, our assumption is false, and line AB must be parallel to line EF, as desired.
Hence, Theorem 3-8 is proved.