Chapter 3.5, Problem 27PPS

To determine

Prove $\overline{AB}\parallel \overline{CD}$

Explanation of Solution

Given:

  $\angle ABC\cong \angle ADC$

  $m\angle A+m\angle ABC=180°$

Calculation :

Statement	Reason
$\angle ABC\cong \angle ADC$	Given
$m\angle A+m\angle ABC=180°$	Given
$m\angle ABC=m\angle ADC$	Definition of congruent angles.
$m\angle A+m\angle ADC=180°$	$m\angle ABC=m\angle ADC$ and $m\angle A+m\angle ABC=180°$
$\overline{AB}\parallel \overline{CD}$	$\angle A\text{ and }\angle ADC$ are supplementary and are consecutive interior angles. Converse of Consecutive Interior Angles Theorem

Converse of Consecutive Interior Angles Theorem:

When a transversal intersects two lines , the two lines are parallel if the consecutive interior angles are supplementary.

Hence proved.