Chapter 9.5, Problem 16WE

To determine

To Prove:that if one pair of opposite sides is congruent of an inscribed quadrilateral, then the other sides is parallel

Explanation of Solution

Given information:

One pair of opposite sides of an inscribed quadrilateral are congruent.

Consider an inscribed quadrilateral ABCD;

  

Since one pair of opposite sides is congruent with the other, $AD=BC$

In ΔADC and ΔBCD

By cyclic quadrilateral

  $\begin{array}{l}m\angle D+m\angle B=180°\mathrm{.........}\left(1\right)\hfill \\ AD=BC\hfill \end{array}$

And $DC=DC$

Since two sides are of the same, then the angle made by these two sides will be same also, therefore,

  $m\angle D=m\angle C\mathrm{.......}(2)$

From equation (1) and (2)

  $m\angle C+m\angle B=180°$

Since the number of interior angles for two parallel lines is 180°,

Therefore,

AB || CD

Therefore, if one pair of opposite sides is congruent, another pair is parallel with an inscribed quadrilateral.