Chapter 12, Problem 40CR

To determine

To prove:QUAD is an isosceles trapezoid.

Explanation of Solution

Given information:

In quadrilateral QUAD,

  $\overline{QU }\cong \overline{AD}$

Angle A is supplementary to angle Q.

  $\overline{QD }\ne \overline{AU}$

Proof:

  

It is given that, In a quadrilateral, if two angles are supplementary, the remaining two angles are also supplementary.

Angle D is supplementary to angle U.

If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

QUAD may be inscribed in a circle.

If chords are congruent, then arcs are also congruent.

  $\stackrel{⌢}{AD}\cong \stackrel{⌢}{QU}$

By reflexive property, we get

  $\stackrel{⌢}{DQ}\cong \stackrel{⌢}{DQ}$

By addition property, we get

  $\stackrel{⌢}{ADQ}\cong \stackrel{⌢}{UQD}$

Angles inscribed in congruent arcs are congruent.

  $\angle A\cong \angle U$

By substitution, we get

  $\angle D$ is supplementary to $\angle A$

If interior angles on same side of traversal are supplementary, then the lines are parallel.

  $\overline{DQ}\parallel \overline{AU}$

A trapezoid is a parallelogram with exactly one pairs of parallel sides.

QUAD is a trapezoid.

If a trapezoid has one pair of congruent sides, it is isosceles.

QUAD is an isosceles trapezoid.