Basic Mathematics for College Students

We have seen that if a quadrilateral has four right angles, it is a rectangle. The
following statements establish some conditions that a parallelogram must meet to
ensure that it is a rectangle.

9.6 Quadrilaterals and Other Polygons 769

Self Check 1

In rectangle shown below,

the length of is 13 ft. Find

each measure:

a.

b.

c.

d.

Now TryProblem 27

m(SG)

m(TG)

m(ST)

m(SRU)

RT

RSTU

Solution
a.In any rectangle, all four angles are right angles. Therefore, is a right
angle, and.

b. and are opposite sides of the rectangle, so they have equal length.
Since the length of is 8 inches, is also 8 inches.
c. and are diagonals of the rectangle, and they intersect at their
midpoints. That means that point is the midpoint of. Since the length of
is 5 inches, is inches, or 10 inches.

d.The diagonals of a rectangle are of equal length. In part c, we found that the
length of WYis 10 inches. Therefore,m(XZ)is also 10 inches.

WA m(WY) 2  5

A WY

WY ZX

WZ m(XY)

XY WZ

m(YXW)90°

YXW

5 ft

12 ft

U

G

RS

T

Parallelograms That Are Rectangles

1. If a parallelogram has one right angle, then the parallelogram is a
   rectangle.


2. If the diagonals of a parallelogram are congruent, then the parallelogram
   is a rectangle.

EXAMPLE (^2) Construction A carpenter wants to build a shed with a
9-foot-by-12-foot base. How can he make sure that the foundation has four right-
angle corners?
Now TryProblem 59
12 ft
12 ft
9 ft 9 ft
AB
DC
StrategyThe carpenter should find the lengths of the diagonals of the
foundation.
WHYIf the diagonals are congruent, then the foundation is rectangular in shape
and the corners are right angles.
SolutionThe four-sided foundation, which we will label as parallelogram ,
has opposite sides of equal length. The carpenter can use a tape measure to find the
lengths of the diagonals and. If these diagonals are of equal length, the
foundation will be a rectangle and have right angles at its four corners. This process
is commonly referred to as “squaring a foundation.” Picture framers use a similar
process to make sure their frames have four 90° corners.

AC BD

ABCD

3 Find unknown angle measures of trapezoids.

A trapezoidis a quadrilateral with exactly two sides parallel. For the trapezoid shown
on the next page, the parallel sides and are called bases.To distinguish
between the two bases, we will refer to as the upper baseand as the lower
base.The angles on either side of the upper base are called upper base angles,and the
angles on either side of the lower base are called lower base angles.The nonparallel
sides are called legs.

AB DC

AB DC