Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems. Prove that a given quadrilateral is a parallelogram.

1. Obj. 26 Parallelograms
The student is able to (I can):
• Prove and apply properties of parallelograms.
• Use properties of parallelograms to solve problems.
• Prove that a given quadrilateral is a parallelogram.

2. parallelogram
A quadrilateral with two pairs of parallel
sides.
A parallelogram has the following
properties:
Opposite sides are parallel. (Definition)
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.

3. 2 pairs of sides
If a quadrilateral has two pairs of parallel sides, then it is a
parallelogram
T
>>
E
I
>>
M
TI ME, TE IM

4. Opp. sides ≅
If a quadrilateral is a parallelogram, then opposite sides are
congruent.
K
G
I
N
KI ≅ NG, GK ≅ IN

5. Opp. angles ≅
If a quadrilateral is a parallelogram, then opposite angles are
congruent.
K
>>
G
O
>>
N
∠K ≅ ∠N, ∠O ≅ ∠G

6. Cons. angles supp.
If a quadrilateral is a parallelogram, then consecutive angles
are supplementary.
1
4
2
3
m∠1 + m∠2 = 180°
m∠2 + m∠3 = 180°
m∠3 + m∠4 = 180°
m∠4 + m∠1 = 180°

7. Diagonals bisect
If a quadrilateral is a parallelogram, then its diagonals
bisect each other.
T
U
>>
S
>>
E
N
TS ≅ NS, ES ≅ US

8. Examples
Find the value of the variable:
1. x =
5x + 3
2x + 15
2. x =
(x + 84)º
3. y =
yº
(3x)º

9. Examples
Find the value of the variable:
1. x = 4
5x + 3
2x + 15
5x + 3 = 2x + 15
3x = 12
2. x = 42
(x + 84)º
3x = x + 84
2x = 84
3. y = 54
3(42) = 126º
y = 180 - 126
yº
(3x)º

10. Conditions for Parallelograms
If one pair of opposite sides of a quadrilateral is congruent
and parallel then the quadrilateral is a parallelogram.
parallel,
We can also use the converses of the theorems from the
previous section to prove that quadrilaterals are
parallelograms.
Parallelogram
⇔
Opposite sides ≅
Opposite angles ≅
Cons. ∠s supp.
Diagonals bisect