Download Geo_Lesson 6_3

Geometry
Lesson 6.3
Proving Quadrilaterals are
Parallelograms
Parallelogram Properties
If a quadrilateral is a parallelogram, then…
• Both pairs of opposite sides are…
Parallel (||)
• Both pairs of opposite sides are…
Congruent ()
• Both pairs of opposite angles are…

Congruent ()
• Consecutive angles are…
Supplementary

• Diagonals have this property…
They bisect each other
• If both pairs of opposite sides of a
quadrilateral are congruent, then it is a
parallelogram
B
C
A
D
• If both pairs of opposite angles of a
quadrilateral are congruent, then it is a
parallelogram
• If an angle of a quadrilateral is
supplementary to both of its consecutive
angles, then it’s a parallelogram
B
(180-x)°
A
x°
(180-x)°
D
C
B
A
C
D
• If the diagonals of a quadrilateral bisect
each other, then it’s a parallelogram
• If one pair of opposite sides of a
quadrilateral are congruent and
parallel, then it’s a parallelogram
B
A
C


D
Example 1
• Can you prove each quad. is a
give a reason in if-then form
a.
b.
c.
Yes!
Yes!
if opp. sides
 & ||,
then 
if diags.
bisect each
other, then 
? If so,
Yes!
if opp. sides
, then 
Practice 1
• Can you prove each quad. is a
give a reason in if-then form
a.
b.
No!
sides
not ||
Parallel sides
would need to
also be
congruent or
congruent sides
would need to
also be parallel

Yes!
? If so,
c.
Yes!

if alt. int. s , if  supp. to
then lines ||
both its consec.
s, then 
if 2 pairs ||
sides, then 
Example / Practice 2
• What additional information do you need
to prove that ABCD is a parallelogram?

• AB || CD AD || BC
• AB  CD
AD  BC
• DCA  BAC DAC  BCA

AE  CE
mCBA + mDAB = 180°
• mCDA + mDAB = 180°
• DE  BE
Example 3
• What values of x and y will make each
quadrilateral into a parallelogram
(a)
A
B
(b)
S
D
C
if opp. sides , then :
3x = 6  x = 2
and (x+2) = (y-1) y = 5
P
Q
R
if opp. s , then 
2x° = 70°  x = 35
and (3x+5)° = (x+3y)°
110°= (35+3y)°
 y = 25
Practice 3
• What values of x and y will make each
quadrilateral into a parallelogram
(a)
A
B
(b)
S
D
C
P
Q
R
rise over _____
run
• The slope of a line is ______
y
• The slope of a line is the change in ____
x
divided by the change in ____
• Two lines are parallel if they have the same
slope
______.
• Two line segments are congruent if they
length
have the same ________.
3 Methods to Prove a Quadrilateral
is a Parallelogram
1. Opposite sides are parallel
Show same slope
2. Opposite sides are congruent. Show same
length
3.One pair of opposite sides  and ||
Show same slope & same length
Example 4: Method 1
• Prove that A(2, -1), B(1, 3), C(6, 5), and
D(7, 1) are the vertices of a parallelogram
y
Method 1:
Slope of AB = -4
Slope of CD = -4
Slope of BC = 2/5
Slope of AD = 2/5
C
B
D
x
A
Because opposite
sides have same
slope, they are ||.
By definition,
ABCD is a
Example 5: Method 2
• Prove that A(2, -1), B(1, 3), C(6, 5), and
D(7, 1) are the vertices of a parallelogram
Method 2:
Length of AB =
Length of CD =
Length of BC =
Length of AD =
y
C
B
D
A
x
17
17
29
29
Because opposite
sides have same
length, they are .
By theorem 6.6,
ABCD is a
Method 3: Opposite Sides are  & ||
• Prove that A(2, -1), B(1, 3), C(6, 5), and
D(7, 1) are the vertices of a parallelogram
Method 3:
Slope of AB = -4
Slope of CD = -4
Length of AB = 17
Length of CD = 17
Because opposite sides
have same slope and
length, they are || &
. By theorem 6.10,
ABCD is a
Practice 1
• Use methods 1 & 2 to prove that M(-4, 7), A(3, 0),
T(2, -5), and H(-5, 2) are the vertices of a parallelogram
M
H
Slopes:
y
A
x
T
Lengths:
MA = -1
MA = 7 2
HT = -1
HT = 7 2
MH = 5
MH =
26
AT = 5
AT =
26
Six ways to show a quadrilateral is a parallelogram:
1.
parallel
Opposite sides are _________
congruent
2. Opposite sides are _________
congruent
3. Opposite angles are _________
supplementary to two
4. One angle is _____________
consecutive angles
bisect each other
5. Diagonals _______
|| and ___

6. One pair of opposite sides is ___
(symbols)
Assignment
• Ch 6.3
(pg. 342-343)
#10-26, #40-46 EVEN