Geometry Notes
Section 6.3 Tests for Parallelograms
There are 6 ways to prove a quadrilateral is a parallelogram – the definition and 5 theorems.
Defn of Parallelogram: a quadrilateral with both pairs of opposite sides parallel
Thm: If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Thm: If one pair of opposite sides of a quadrilateral are parallel and congruent,
then the quadrilateral is a parallelogram.
Thm: If both pairs of opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Thm: If the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram.
Thm: If one angle of a quadrilateral is supplementary
to both of its consecutive angles,
then the quadrilateral is a parallelogram.
Ex: Determine whether the quadrilateral is a parallelogram. Justify your answer with a definition
or theorem.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Coordinate Geometry
Slope Formula:
y y
m 2 1
x2  x1
Given 2 points  x1 , y1  and  x2 , y2 
Distance Formula:
d
 x2  x1    y2  y1 
2
2
Midpoint Formula:
 x  x y  y2 
M  1 2 , 1

2 
 2
 To show sides are ||, show their _____________________ are equal.
 To show sides are , show their _____________________ are equal.
Ex: Determine whether quadrilateral JKLM is a parallelogram using the definition of parallelogram.
J(–1, –6), K(–4, –1), L(4, 5), M(7, 0)
L
K
M
Both pairs of opposite sides are ||, so JKLM is a parallelogram by defn of .
J
Ex: Show that FGHJ is a parallelogram using the distance and slope formulas to
show one pair of opposite sides are both parallel and congruent.
F(‒4, ‒2), G(‒2, 2), H(4, 3), J(2, ‒1)
Ex: Show that KLMN is a parallelogram using the midpoint formula to
show the diagonals bisect each other.
K(‒3, 0), L(‒5, 7), M(3, 5), N(5, ‒2)
Assignment: Sec. 6.3 p.417 # 1, 2, 4 – 7, 9 – 14, 18 – 25
(20 probs)