Chapter 3.5, Problem 31PPS

To determine

Prove Alternate Interior Angle Converse.

Explanation of Solution

Given:

Alternate Interior Angle Converse :

If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.

Calculation :

Consider the diagram :

  

Given :

  $\angle 8\cong \angle 6$

To Prove : $p\parallel q$

Proof:

  $\angle 8\cong \angle 3$ …………[Vertically opposite angles]

Now, $\angle 8\cong \angle 6$ and $\angle 8\cong \angle 3$

  $\Rightarrow \angle 6\cong \angle 3$ ………..[Transitive Property]

And $\angle 6$ and $\angle 3$ are corresponding angles .

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 p and q are parallel.

Hence proved.