Chapter 5.2, Problem 12WE

To determine

To proof: that quadrilateral is a parallelogram with the help of given information.

Answer to Problem 12WE

The quadrilateral $\square ABCD$ is a parallelogram.

Explanation of Solution

Given information:

Theorem: If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

Given: $ABCD$ is a quadrilateral.

Calculation:

  $AB\parallel BC$

  $AB\cong BC$

To Prove: $\square ABCD$ is a parallelogram

The diagram is shown below:

  

In $△ADC$ and $△ABC$

  $AB\parallel BC\dots \dots$ (1)

Thus,

  $AB=BC$

  $\angle DCA=\angle BAC$ (Alternate interior angle)

  $AC=CA$ (Common side)

By angle side angle property;

  $△ADC\cong △ABC$

By corresponding parts of congruent triangles

  $\angle DAC\cong \angle BCA$

But this is alternate interior angle in $\text{Δ}DAC$ and $△BCA$ respectively.

Then

  $AD\parallel BC\dots \dots$ (2)

From equation (1) and (2) opposite side is parallel.

Thus $\square ABCD$ is a parallelogram.