Geometry Notes
6-3 Conditions for Parallelograms
In Section 6-2, we looked at the definition of a parallelogram and special properties of
parallelograms. In this section, we will work backwards. Given a quadrilateral, how do we
know whether or not it is a parallelogram? There are several ways to tell.
Conditions for Parallelograms
A quadrilateral is a parallelogram if:
1) Both pairs of opposite sides are parallel (remember this is the definition.)
2) Both pairs of opposite sides are congruent.
3) Both pairs of opposite angles are congruent.
4) One pair of opposite sides is both parallel AND congruent.
5) An angle is supplementary to both of its consecutive angles.
6) Both diagonals bisect each other.
Ex. 1: Determine if each of the following must be a parallelogram. If so, give the reason.
a)
b)
115°
°
65°
°
65°
°
Yes, one pair of sides parallel and ≅
Yes, one angle is supp. to both
consecutive angles
c)
d)
No
Technically, yes. You can show that
the parallel sides are ≅ by ASA.
e)
f)
6
8
8
6
Yes, opposite angles are ≅.
Yes, diagonals bisect each other.
g)
h)
No
Yes, both pairs of opposite sides
are parallel.
Ex. 2: Show that JKLM is a parallelogram if
a = 3 and b = 9.
JK = 15a −11 = 15(3) −11 = 45 −11 = 34
LM = 10a + 4 = 10(3) + 4 = 34
KL = 5b + 6 = 5(9) + 6 = 45 + 6 = 51
JM = 8b − 21 = 8(9) − 21 = 72 − 21 = 51
Opposite sides are congruent, so it’s a parallelogram.
Ex. 3: Show that PQRS is a parallelogram if
x = 10 and y = 6.5.
m∠R = 15x −16 = 15(10) −16 = 150 −16 = 134°
m∠Q = 6y + 7 = 6(6.5) + 7 = 39 + 7 = 46°
m∠S = 8y − 6 = 8(6.5) − 6 = 52 − 6 = 46°
Angle R is supplementary to both consecutive angles (Q and S), so it’s a parallelogram.