Chapter 3.5, Problem 30PPS

To determine

Prove Perpendicular Transversal Converse Theorem.

Explanation of Solution

Given:

Perpendicular Transversal Converse Theorem:

In a plane , if two lines are perpendicular to the same line , then they are parallel.

Calculation :

Consider the diagram :

  

Given :

  $l\perp t$ and $m\perp t$

To Prove : $l\parallel m$

Since $l\perp t$ and $m\perp t$ , $\angle 1$ and $\angle 2$ are $90°$ .

Since

  $\begin{array}{l}m\angle 1=m\angle 2\\ \Rightarrow \angle 1\cong \angle 2\end{array}$

And $\angle 1$ and $\angle 2$ are corresponding angles where t is the transversal to the lines l and m .

By the Converse of Corresponding Angle Theorem which states that when a transversal intersects two lines , the two lines are parallel if the corresponding angles are congruent, the lines l and m are parallel.

Hence proved.