Write a paragraph proof of this theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other
Sure! Here is a paragraph proof of the theorem you requested:
Given: Line l and lines m and n are perpendicular to line l.
To Prove: Lines m and n are parallel to each other.
Proof:
1. Let's assume lines m and n intersect at point P.
2. Since line m is perpendicular to line l, the angle formed between them is a right angle.
3. Similarly, since line n is also perpendicular to line l, the angle formed between them is also a right angle.
4. By definition of perpendicular lines, if two lines are perpendicular to the same line, then the angles formed between those lines and the given line are congruent.
5. Therefore, the right angle formed between line m and line l is congruent to the right angle formed between line n and line l.
6. By definition of parallel lines, two lines are parallel if their corresponding angles are congruent.
7. Hence, by step 5 and 6, lines m and n are parallel to each other.
Therefore, we have proved that in a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
To prove the theorem "In a plane, if two lines are perpendicular to the same line, then they are parallel to each other," we can use a paragraph proof.
Let's assume we have two lines, line AB and line CD, that are perpendicular to line EF. Our goal is to show that line AB is also parallel to line CD.
When two lines are perpendicular to the same line, it implies that they are each perpendicular to any line parallel to the common line.
Given that line AB is perpendicular to line EF, we can conclude that line AB is also perpendicular to any line parallel to EF.
Now, consider line CD. We know that line CD is perpendicular to line EF since it has also been specified in the problem statement. Therefore, by the transitive property of perpendicular lines, line CD is perpendicular to any line parallel to EF as well.
Since both line AB and line CD are perpendicular to any line parallel to EF, we can conclude that line AB and line CD are parallel to each other by definition.
Hence, we have proven that in a plane, if two lines are perpendicular to the same line, then they are parallel to each other.