Chapter 5.7, Problem 7PSA

To determine

To prove:ABCDis a trapezoid.

Explanation of Solution

Given information:

  $\overline{AC }$ bisectsangle BAD.

  $\overline{AB }\cong \overline{BC }$

  $\overline{AB }$ is not parallel to $\overline{CD }$ .

Proof:

  

It is given that,

  $\overline{AB }\cong \overline{BC }$

If corresponding sides are congruent then corresponding angles are congruent.

  $\angle BAC\cong \angle BCA$

It is given that,

  $\overline{AC }$ bisectsangle BAD.

An angle bisector divides the angle into two angles with equal measures.

By definition of bisector, we get

  $\angle BAC\cong \angle CAD$

By transitive property, we get

  $\angle BCA\cong \angle CAD$

If alternate interior angles are congruent then lines are parallel.

  $\overline{BC }\parallel \overline{AD}$

It is given that,

  $\overline{AB }$ is not parallel to $\overline{CD }$ .

If one pair of sides are parallel, then figure is trapezoid.

ABCD is a trapezoid.