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8.4 Rectangles
Objectives

Recognize and apply properties of rectangles

Determine whether parallelograms are
rectangles
Rectangles

A rectangle is a parallelogram with four
right angles.
Rectangles

Since rectangles are parallelograms, they
have all their properties:





Opposite sides are || and ≅.
Opposite s are ≅.
Consecutive s are supplementary.
Diagonals bisect each other.
In addition, there exists Theorem 8.13
which states if a
is a rectangle then the
diagonals are ≅.
Example 1:
Quadrilateral RSTU is a rectangle. If
find x.
and
Example 1:
The diagonals of a rectangle are congruent,
Definition of congruent segments
Substitution
Subtract 6x from each side.
Add 4 to each side.
Answer: 8
Your Turn:
Quadrilateral EFGH is a rectangle. If
find x.
Answer: 5
and
Example 2a:
Quadrilateral LMNP is a rectangle. Find x.
Example 2a:
Angle Addition Theorem
Substitution
Simplify.
Subtract 10 from each side.
Divide each side by 8.
Answer: 10
Example 2b:
Quadrilateral LMNP is a rectangle. Find y.
Example 2b:
Since a rectangle is a parallelogram, opposite sides are
parallel. So, alternate interior angles are congruent.
Alternate Interior Angles Theorem
Substitution
Simplify.
Subtract 2 from each side.
Divide each side by 6.
Answer: 5
Your Turn:
Quadrilateral EFGH is a rectangle.
a. Find x.
b. Find y.
Answer: 7
Answer: 11
Example 3:
Kyle is building a barn for his horse. He measures the
diagonals of the door opening to make sure that they
bisect each other and they are congruent. How does
he know that the corners are
angles?
Answer: We know that
A parallelogram with
congruent diagonals is a rectangle. Therefore,
the corners are
angles.
Your Turn:
Max is building a swimming
pool in his backyard. He
measures the length and
width of the pool so that
opposite sides are parallel. He
also measures the diagonals
of the pool to make sure that
they are congruent. How does
he know that the measure of
each corner is 90?
Answer: Since opposite sides are parallel, we know that
RSTU is a parallelogram. We know that
.
A parallelogram with congruent diagonals is a
rectangle. Therefore, the corners are
Example 4:
Quadrilateral ABCD has vertices A(–2, 1), B(4, 3), C(5, 0),
and D(–1, –2). Determine whether ABCD is a rectangle
using the Slope Formula.
Method 1: Use the Slope Formula,
to see if
consecutive sides are perpendicular.
Example 4:
quadrilateral ABCD is a
parallelogram. The product of the slopes of consecutive
sides is –1. This means that
Answer: The perpendicular segments create four right
angles. Therefore, by definition ABCD is a
rectangle.
Example 4:
Method 2: Use the Distance Formula,
to determine whether
opposite sides are congruent.
Example 4:
Since each pair of opposite sides of the quadrilateral have
the same measure, they are congruent. Quadrilateral
ABCD is a parallelogram.
Example 4:
Find the length of the diagonals.
The length of each diagonal is
Answer: Since the diagonals are congruent, ABCD is a
rectangle.
Your Turn:
Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3),
Y(3, 1), and Z(2, –1). Determine whether WXYZ is a
rectangle using the Distance Formula.
Your Turn:
Answer:
we
can conclude that opposite sides of the
quadrilateral are congruent. Therefore, WXYZ is
a parallelogram. Diagonals WY and XZ each
have a length of 5. Since the diagonals are
congruent, WXYZ is a rectangle by Theorem
8.14.
Assignment

Pre-AP Geometry
Pg. 428 #10 - 32, 36, 42

Geometry:
Pg. 428 #4-7, 10 - 26