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.
Given: \( R \perp S \), \( T \perp S \)
Prove: \( R \parallel T \)
Proof: Since line \( R \) is perpendicular to line \( S \), and line \( T \) is also perpendicular to line \( S \), both lines \( R \) and \( T \) form right angles with line \( S \). According to the definition of perpendicular lines, both \( R \) and \( T \) create a \(90^\circ\) angle with line \( S \). By the converse of the Alternate Interior Angles Theorem, if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. In this case, line \( S \) acts as the transversal cutting lines \( R \) and \( T \). The angles formed are right angles, which are congruent. Therefore, lines \( R \) and \( T \) are parallel to each other. Hence, \( R \parallel T \).
Answer (2)
If you call the gradient of S m:
Gradient of R if perpendicular to S = -1/m
Gradient of T if perpendicular to S = -1/m
-1/m=-1/m, so the lines are parallel
The theorem states that if two lines are perpendicular to the same line, they are parallel. By showing that both lines create right angles with the same transversal, we apply the Alternate Interior Angles Theorem. Thus, lines R and T must be parallel to each other, as stated.
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