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8.3 Test for Parallelograms
Check.3.2 Connect coordinate geometry to geometric
figures in the plane (e.g. midpoints, distance formula, slope,
and polygons).
Spi.3.2 Use coordinate geometry to prove characteristics of
polygonal figures.
Check.4.19 Use coordinate geometry to prove properties of
plane figures..
Objective: Be able to identify parallelogram.
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
A
B
C
D
Properties of Parallelogram – Converse Statements
Opposite Sides of a parallelogram are congruent
Opposite Angles of a parallelogram are congruent
Consecutive Angles of a parallelogram are supplementary.
Diagonals of a parallelogram bisect each other
Proving Parallelograms
A
C
B
D
If opposite sides of a quadrilateral are parallel,
then it is a parallelogram
Proving Parallelograms
A
C
B
D
If opposite sides of a quadrilateral are congruent,
then it is a parallelogram
Parallelograms
A
C
B
D
If both pairs of opposite angles of a quadrilateral are
congruent
then it is a parallelogram
Parallelograms
A
C
B
D
If diagonals of a quadrilateral bisect each other,
then
then it is a parallelogram
Proving Parallelograms
A
C
B
D
If one pair of opposite sides of a quadrilateral is
both parallel and congruent,
then it is a parallelogram
Proving Parallelograms
Given AC & BD
If both pairs of opposite angles of a
quadrilateral are congruent then it is
a parallelogram
CPCTC,
If opposite sides of a
quadrilateral are
congruent, then it is a
parallelogram
Proving Parallelograms
If both pairs of opposite angles of a
quadrilateral are congruent then it is a
parallelogram
If opposite sides of a quadrilateral are
congruent, then it is a parallelogram
Using Properties
Find x and y so that the quadrilateral is
a parallelogram.
If opposite sides of a quadrilateral are
congruent, then it is a parallelogram
4y = 6y - 42
42 = 6y – 4y
42 = 2y
y = 21
6x - 12 = 2x + 36
6x – 2x = 36 + 12
4x = 48
x = 12
Using Properties
Find x and y so that the quadrilateral is
a parallelogram.
If diagonals of a quadrilateral bisect
each other, then , then it is a
parallelogram
x = 5x - 28
28 = 4x
2y + 12 = 5y
12 = 3y
y=4
x=7
Using Slope to prove Parallelograms
If opposite sides of a quadrilateral are
parallel then it is a parallelogram.
If two lines are parallel, they have the
same slope.
BC
AB
AD
DC
Both sides have the same slope,
therefore they are parallel and it is a
parallelogram
Using Slope and Distance
to prove Parallelograms
Is PS  QR and is PS parallel with QR?
Since one pair of opposite sides is congruent
and parallel, then PQRS is a parallelogram.
Summary – Test for Parallelograms
1.
2.
3.
4.
5.
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent.
Both pairs of opposite angles are congruent.
Diagonals Bisect each other
A pair of opposite sides are both parallel and
congruent.
• Practice Assignment
– Page 414 10 -24 Even