15-3: Parallelograms
Objectives:
1. To discover and use
properties of
parallelograms
2. To find the area of
parallelograms
Assignment:
SpringBoard:
• P. 215: 11, 12
• P. 221: 5-7
Purple Geometry Book:
• P. 723-724: 1, 3, 4,
10, 11, 17, 23, 24, 26
• Challenge Problems
Objective 1
You will be able to discover and
use properties of parallelograms
Parallelograms
What makes a polygon a parallelogram?
Parallelogram
A parallelogram is a
quadrilateral with
both pairs of
opposite sides
parallel.
Written
𝑃𝑄𝑅𝑆
𝑃𝑄 ∥ 𝑅𝑆
and
𝑄𝑅 ∥ 𝑃𝑆
The Story of Parallelograms
For this lesson, we’ll be writing The Story of
Parallelograms. While short on plot, the
story is definitive and is illustrated. In
order to write this book, we’ll need to make
a small, 8-page booklet with no staples.
That part is magic.
The Story of Parallelograms: Page Layout
1. Title
2. Definition
3. Picture
4. Theorem 1
5. Theorem 2
6. Theorem 3
7. Theorem 4
8. Area
Investigation 1
In this Investigation, we
will be using
Geogebra to construct
a perfect
parallelogram, and
then we will discover
four useful properties
about parallelograms.
Theorem 1
If a quadrilateral is a
parallelogram, then
its opposite sides
are congruent.
If PQRS is a parallelogram, then PQ  RS and QR  PS.
Theorem 2
If a quadrilateral is a
parallelogram, then
its opposite angles
are congruent.
If PQRS is a parallelogram, then P  R and Q  S .
Theorem 3
If a quadrilateral is a
parallelogram, then
consecutive angles
are supplementary.
If PQRS is a parallelogram, then x + y = 180°.
Theorem 4
If a quadrilateral is a
parallelogram, then
its diagonals bisect
each other.
Construction
Use your compass and straightedge along with the
properties of parallelograms to construct this
peculiar quadrilateral.
Example 1
Find each indicated measure.
1. NM
2. KM
3. mJKL
4. mLKM
Example 2
The diagonals of
parallelogram
LMNO
intersect at
point P. What
are the
coordinates of
P?
Example 3
Find the values of c and d.
Example 4
For the parallelogram below, find the values
of t and v.
Example 5: SAT
For parallelogram ABCD, if AB > BD, which
of the following statements must be true?
I. CD < BD
II. ADB > C
III. CBD > A
B
A
C
D
Example 6
Prove: If a quadrilateral is a parallelogram,
then the diagonals bisect each other
Given:
Prove:
You will be able to find the
area of parallelograms
Bases and Heights
Any one of the sides of a parallelogram can
be considered a base. But the height of a
parallelogram is not necessarily the length
of a side.
Bases and Heights
The altitude is any segment from one side of
the parallelogram perpendicular to a line
through the opposite side. The length of
the altitude is the height.
Bases and Heights 2
The altitude is any segment from one side of
the parallelogram perpendicular to a line
through the opposite side. The length of
the altitude is the height.
Investigation 2
Now you will discover
a formula for
computing the area
of a parallelogram.
Area of a Parallelogram Theorem
The area of a parallelogram is the product of
a base and its corresponding height.
A = bh
Height (h)
Height (h)
Base (b)
Base (b)
Area of a Parallelogram Theorem
The area of a parallelogram is the product of
a base and its corresponding height.
A = bh
Example 7
Find the area of parallelogram PQRS.
Example 8
What is the height of
a parallelogram that
has an area of 7.13
m2 and a base 2.3
m long?
15-3: Parallelograms
Objectives:
1. To discover and use
properties of
parallelograms
2. To find the area of
parallelograms
Assignment:
SpringBoard:
• P. 215: 11, 12
• P. 221: 5-7
Purple Geometry Book:
• P. 723-724: 1, 3, 4,
10, 11, 17, 23, 24, 26
• Challenge Problems