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Geometry
Lesson 6.2
Properties of Parallelograms
Review of parallel lines
1
3 2


• What type of angles are 1 and 2?
alternate interior
• What type of angles are 1 and 3?
consecutive interior
50°
• If m1 = 50°, then m2 = _____
130°
• If m1 = 50°, then m3 = _____
• A special type of quadrilateral
• Both pairs of opposite sides are
parallel
Q
P
R


S
PQ || RS
PS || QR
PQRS is called “parallelogram PQRS”
Applications of Parallelograms:
Car Suspension
4-bar Coil Spring
4-bar linkage
http://www.carbibles.com/suspension_bible.html
Applications of Parallelograms: Lifts
A simple
mechanism for
raising a heavy
load for
maintenance
Applications of Parallelograms:
Binocular Mount (Astronomy)
• Holds binoculars steady for night sky
viewing
• Allows full range of motion around the
sky
• If a quadrilateral is a parallelogram,
then its opposite sides are congruent
Q
P
R
S
• If a quadrilateral is a parallelogram,
then its opposite angles are congruent
• If a quadrilateral is a parallelogram, then
its consecutive angles are supplementary
mQ + mR = 180°
Q
P
QM  SM and PM  RM
Q
R
S
P
R
M
S
• If a quadrilateral is a parallelogram, then
its diagonals bisect each other
Example 1
• Given: MNOP is a parallelogram
OP and MN  ____
OP
• MN || ____
MP
• ON  ____

MNO
• MPO  ______
NQ
• PQ  _____
OQ
• MQ  _____
PQO
• MQN  ______
MNP
• NPO  ______

Practice 1
• Find the measures in parallelogram HIJK
• HI = 16
• KH = 10
• GH = 8
• HJ = 16
• mKIH = 28°
• mJIH = 96°
• mKJI = 84°
• mHKI = 68°
16
8
10
28°
68°
84°
Example 2
• Find the value of x, y, and z
Recognize: AD = BC
and AB = CD
AD = BC
3x – 7 = 2x + 9
x = 16
AB = CD
5y = 2y + 18
3y = 18
y = 6
B
C
A
120°
3z°
D
mA + mD = 180°
120° + 3z° = 180°
3z° = 60°
z = 20
Practice 2 Find x, y, and z
(b)


(a)


5x = 3x + 18
x = 9
3y – 7 = 2y + 4
y = 11
(2x+12)° = (3x–18)° x = 30
4y°+ (3x – 18)° = 180°
4y°+ 72° = 180°  y = 27
3z° = 4y° = 108°  z = 36
Practice 2, cont.
(c)

• Find x and y
(d)


3x + 6 = 12
x = 2
3x + 2 = 23  x = 7
2y + 9 = 27
y = 9
7y – 2 = 5y + 4
2y = 6  y = 3

Example 3: Proofs Using Properties of
Statement
ABCD is a
BD  BD
BC  AD, AB  CD
ABD  CDB
Reason
Given
Plan:
Use opp. sides
theorem and SSS
if AB exists, then AB  AB
if , then opp. sides 
if 3 sides , then s  (SSS)
Practice 3: Proofs Using Properties of
Plan:
Use opp. sides
theorem and SSS
Statement
ABCD is a
AD  CB
Reason
Given
if , then opp. sides 
AE  CE, BE  DE if , then diags. bisect each other
AED  CEB
if 3 sides , then s  (SSS)
Name five properties of parallelograms
and label the diagram with each:
Parallel (by definition)
1. Opposite sides are __________
Congruent
2. Opposite sides are __________
Congruent
3. Opposite angles are _____________
Supplementary
4. Consecutive angles are ___________
bisect each other
5. Diagonals ______
Assignment
Ch 6.2 (pg. 333-334)
#2-36 EVEN