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.

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

To prove Theorem 3-8, "In a plane, if two lines are perpendicular to the same line, then they are parallel to each other," we can use a proof by contradiction. Suppose that lines m and n are both perpendicular to line l, and that lines m and n are not parallel to each other. This means that there exists a point A on line m and a point B on line n such that AB is not parallel to line l. Let's draw the diagram to better visualize this scenario. Now, because line m is perpendicular to line l, we can form a right angle at point A. Similarly, because line n is perpendicular to line l, we can form a right angle at point B. This means that angle A and angle B are both right angles. However, the sum of angles in a triangle is always 180 degrees. Therefore, the sum of angle A and angle B is 180 degrees, which contradicts the fact that they are both right angles. Hence, our assumption that lines m and n are not parallel to each other must be false. Therefore, we can conclude that if two lines are perpendicular to the same line, they are parallel to each other.