Exercise 50 Page 937

We are given a design of a stained glass window. We want to know what kind of figure the quadrilateral $ACEG$ is. First, let's review the classification of quadrilaterals.

Quadrilateral	Definition
Parallelogram	Both pairs of opposite sides are parallel.
Rhombus	Parallelogram with four congruent sides.
Rectangle	Parallelogram with four right angles.
Isosceles trapezoid	Trapezoid with legs that are congruent.
Kite	Quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.
Square	Parallelogram with four congruent sides and four right angles.

Now we will take a look at the quadrilateral $ACEG.$

We can see that every angle of the quadrilateral intercepts a semicircle. Since the measure of a semicircle is $180^{\circ },$ we know by the Inscribed Angle Theorem that the measure of each angle of the quadrilateral is $90^{\circ }.$ Therefore, each angle of our quadrilateral is a right angle. Note that each side of our quadrilateral is a chord of an arc that is comprised of $2$ small arcs. The small arcs are congruent, so the arcs made of $2$ small arcs are congruent as well. This means that the chords of these arcs, which are the sides of our quadrilateral, are congruent.

Both pairs of opposite sides of our quadrilateral are congruent, so the quadrilateral is a parallelogram. Additionally, all the angles in our quadrilateral are right angles and all its sides are congruent, so the quadrilateral is a square.