Download Properties of Parallelograms

PROPERTIES OF
PARALLELOGRAMS
Geometry CP2 (Holt 6-2)
K.Santos
QUADRILATERAL
Quadrilateral—is a polygon with four sides
PARALLELOGRAM
Parallelogram—a quadrilateral with two pairs of
parallel sides
A
B
𝐴𝐵⃒⃒ 𝐷𝐶
𝐴𝐷⃒⃒ 𝐵𝐶
D
C
Parallelogram ABCD
ABCD
THEOREM 6-2-1
Parallelogram----opposite sides of are
congruent.
A
B
Type
equation
here.
D
𝐴𝐵 ≅ 𝐷𝐶
𝐴𝐷 ≅ 𝐵𝐶
C
THEOREM 6-2-2
Parallelogram---opposite angles are
congruent.
W
X
Z
Y
<W≅<Y
<X ≅<Z
THEOREM 6-2-3
Parallelogram---consecutive angles are
supplementary.
A
B
D
m<A + m<B = 180°
m<B + m<C = 180°
m<C + m<D = 180°
m<D + m<A = 180°
C
EXAMPLE--ANGLES
A
B
Find the value of x in
ABCD. Then find the measures
of all the angles. m< B = x +15° and m<D= 135 - x°
D
<B≅<D
x + 15 = 135- x
2x + 15 = 135
2x = 120
x = 60°
m < B = x + 15 = 60 + 15 = 75°
m < D = 75°
< A is supplementary to < B
So 180 – 75 = 105°
m < A = 105° and m < C = 105°
C
THEOREM 6-2-4
Parallelogram---diagonals of bisect each
other.
M
N
X
P
O
𝑀𝑂 and 𝑁𝑃 bisect each other
𝑀𝑋 ≅ 𝑂𝑋
𝑁𝑋 ≅ 𝑃𝑋
EXAMPLE--ANGLES
Given parallelogram ABCD with m<A = 70°. Find
the measures of the other angles.
A
B
D
C
<A and <C are opposite angles---congruent
m<C = 70°
<A and <B are consecutive angles---supplementary
m<B = 180 -70 = 110°
<B and <D are opposite angles—congruent
m<D = 110°
EXAMPLE—DIAGONALS
Given the parallelogram below. Find w and z.
10
3w
18
2z
Diagonals of a parallelogram bisect each other.
3w = 18
2z = 10
w=6
z=5
PROPERTIES OF PARALLELOGRAMS
--both pairs of opposite sides are parallel
--both pairs of opposite sides are congruent
--both pairs of opposite angles are congruent
--all pairs of consecutive angles are supplementary
--both diagonals bisect each other