Transcript 6.2 Properties of Parallelograms

Properties of Parallelograms
Unit 12, Day 1
From the presentation by
Mrs. Spitz, Spring 2005
http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.2%20Parallelograms.ppt
You will need:
•
•
•
•
•
Index card
Scissors
1 piece of tape
Ruler
Protractor
Exploration
1. Mark a point
somewhere along the
bottom edge of your
paper.
2. Draw a line from that
point to the top right
corner of the
rectangle to form a
triangle.
Amy King
Exploration
3. Cut along this line to
remove the triangle.
4. Attach the triangle to
the left side of the
rectangle.
5. What shape have you
created?
Amy King
Q
R
In this lesson . . .
P
S
And the rest of the unit, you will study special
quadrilaterals. A parallelogram is a quadrilateral
with both pairs of opposite sides parallel.
When you mark diagrams of quadrilaterals, use
matching arrowheads to indicate which sides are
parallel. For example, in the diagram above,
PQ║RS and QR║SP. The symbol
PQRS is
read “parallelogram PQRS.”
Exploration
Measure the lengths of the
sides of your
parallelogram.
What conjecture could
you make regarding the
lengths of the sides of a
parallelogram?
Amy King
Theorems about parallelograms
Q
• 9-1—If a
quadrilateral is a
parallelogram,
then its opposite
sides are
congruent.
►PQ≅RS and
SP≅QR
P
R
S
Exploration
Measure the angles of
your parallelogram.
What conjecture could
you make regarding the
angles of a parallelogram?
Amy King
Theorems about parallelograms
Q
R
• 9-2—If a
quadrilateral is a
parallelogram,
then its opposite
angles are
congruent.
P ≅ R and
Q ≅ S
P
S
Theorems about parallelograms
Q
• If a quadrilateral is a
parallelogram, then its
consecutive angles are
supplementary (add
up to 180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
mS + mP = 180°
P
R
S
Exploration
Draw both of the
diagonals of your
parallelogram.
Measure the distance
from each corner to the
point where the
diagonals intersect.
Amy King
Exploration
What conjecture could
you make regarding the
lengths of the diagonals of
a parallelogram?
Amy King
Theorems about parallelograms
Q
• 9-3—If a
quadrilateral is a
parallelogram,
then its diagonals
bisect each other.
QM ≅ SM and
PM ≅ RM
R
M
P
S
Ex. 1: Using properties of
Parallelograms
•
FGHJ is a
parallelogram. Find
the unknown length.
a. JH
b. JK
5
F
G
K
J
b.
3
H
Ex. 1: Using properties of
Parallelograms
5
F
SOLUTION:
a. JH = FG so JH = 5
G
K
J
b.
3
H
Ex. 1: Using properties of
Parallelograms
5
F
SOLUTION:
b. JK = GK, so JK = 3
G
K
J
b.
3
H
Ex. 2: Using properties of parallelograms
Q
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
P
R
70°
S
Ex. 2: Using properties of parallelograms
Q
R
a. mR = mP , so mR = 70°
70°
P
S
Ex. 2: Using properties of parallelograms
Q
R
b. mQ + mP = 180°
mQ + 70° = 180°
mQ = 110°
P
70°
S
Ex. 3: Using Algebra with Parallelograms
P
PQRS is a parallelogram.
Find the value of x.
mS + mR = 180°
3x + 120 = 180
3x = 60
x = 20
S
3x°
120°
R
Q
Ex. 4: Using parallelograms in real life
FURNITURE DESIGN. A drafting
table is made so that the legs can be
joined in different ways to change
the slope of the drawing surface.
In the arrangement below, the legs
AC and BD do not bisect each
other. Is ABCD a parallelogram?
C
B
A
D
Ex. 4: Using parallelograms in real life
FURNITURE DESIGN. A drafting
table is made so that the legs can be
joined in different ways to change
the slope of the drawing surface.
In the arrangement below, the legs
AC and BD do not bisect each
other. Is ABCD a parallelogram?
ANSWER: NO. If ABCD were a
parallelogram, then by Theorem
6.5, AC would bisect BD and BD
would bisect AC. They do not, so
it cannot be a parallelogram.
C
B
A
D
Homework
Work Packets: Properties of
Parallelograms