Chapter 3.5, Problem 23PPS

To determine

Prove $l\parallel m$

Explanation of Solution

Given:

  $\angle 1\text{ and }\angle \text{2}$ are supplementary

  

Calculation :

Given: $\angle 1\text{ and }\angle \text{2}$ are supplementary.

  $\angle 2\text{ and }\angle 3$ form a linear pair ……….[definition of linear pair]

  $\angle 2\text{ and }\angle 3$ are supplementary………[definition of linear pair…their sum is $180°$ ]

So,

  $\begin{array}{l}m\angle 1+m\angle 2=180°=m\angle 2+m\angle 3\\ \Rightarrow m\angle 1+m\angle 2=m\angle 2+m\angle 3\\ \Rightarrow m\angle 1=m\angle \mathrm{3...................}[\text{subtract }m\angle 2\text{ from each side]}\\ \Rightarrow \angle \text{1}\cong \angle \text{3}\end{array}$

Converse of Corresponding Angles Postulate :

When a transversal intersects two lines , the two lines are parallel if the corresponding angles are congruent.

So, by the converse of Corresponding Angles Postulate, $l\parallel m$ .

Hence proved.