Download a parallelogram is a quadrilateral in which both pairs of opposite

Class Notes
Section 4.1
Parallelograms
Defn – a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Thm – A diagonal of a parallelogram separates it into two congruent triangles.
Proof:
Corollary – The opposite angles of a parallelogram are congruent.
Corollary – The opposite sides of a parallelogram are congruent.
Corollary – The diagonals of a parallelogram bisect each other.
Corollary – Two consecutive angles of a parallelogram are supplementary.
Thm – Two parallel lines are everywhere equidistant.
Defn – An altitude of a parallelogram is a line segment from one vertex that is perpendicular to a
non adjacent side (or an extension of that side).
Thm – in a parallelogram with unequal pairs of consecutive angles, the longer diagonal lies opposite
the obtuse angle.
As seen in the flow chart below, a rectangle, a rhombus, and a square are all parallelograms.
Quadrilateral
No special properties
One pair of parallel sides
2 pairs of parallel sides

^

Parallelogram
^
Trapezoid


4 congruent sides
4 right angles
Rectangle
Rhombus
4 right angles and 4 congruent sides
Square
Use for Popper 10 question 1 and question 2:
E

4x + 2°


A. 10°
B. 138°
Popper 10 question 2: Find
A. 10°
B. 138°
∠
G
in parallelogram EFGH.
C. 34°
∠
F

H
Popper 10 question 1: Find
x + 8°
D. 42°
D. None of these
in parallelogram EFGH.
C. 34°
D. 42°
D. None of these
“RULES” of parallelograms:
1)
2)
3)
4)
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles in a parallelogram are supplementary.
The diagonals of a parallelogram bisect each other.
Example 1:
MATH is a parallelogram. Find the values of w, x, y, and z.
M
55
15
y
w
z
x
H
A
T
Example 2:
If 1  3 and 2  4, is quadrilateral ABCD a parallelogram?
A
1
B
2
4
D
3
C
Example 3:
Find the measure of each angle and side in parallelogram ABCD below.
A
2y + 1
2x
8
x
D
y+6
C
B
Example 4 :
B
A
Diagonals
measure of:
2x
R
=
14
D
C
Example 5:
Find the measure of each angle in parallelogram ABCD below.
A
28
35
47
D
C
Example 6: Find
x = ________
y = ________
y
4
x+y
2x - 2
B
and
intersect at R. Find the
Methods might be used to prove that a quadrilateral is a parallelogram.
1. If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a
parallelogram.
2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
3. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the
quadrilateral is a parallelogram.
4. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
5. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Example 7: State whether or not you can conclude that the figure is a parallelogram, based on the
given information.
a.
AB CD and AD BC
b.
AB || CD and AD || BC
c.
AB CD and AB || CD
d.
AD BC and AB || CD
e.
AE  AC and BE BD
f.
g.
AB BC CD AD
mADC mABC and mBAD mBCD
A
B
E
C
D
Popper 10 question 3: Which of the following is NOT a quadrilateral?
A. rhombus
B. parallelogram
C. triangle
D. square
Popper 10 question 4: Which quadrilateral has four (4) congruent sides?
A. rhombus
B. trapezoid
C. rectangle
D. parallelogram
Popper 10 question 5: Which of the following has only one (1) pair of parallel sides?
A. square
B. rectangle
C. parallelogram
D. trapezoid