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
“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
H
15
y
w
x
z
T
A
Example 2:
If 1  3 and 2  4, is quadrilateral ABCD a parallelogram?
A
B
2
1
4
D
3
C
Example 3:
Find the measure of each angle and side in parallelogram ABCD below.
A
2y + 1
B
2x
8
x
y+6
D
C
Example 4 :
B
A
Diagonals
measure of:
2x
R
D
=
14
C
Example 5:
Find the measure of each angle in parallelogram ABCD below.
A
35
D
28
47
C
B
and
intersect at R. Find the
Example 6: Find
x = ________
y = ________
y
4
x+y
2x - 2
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.
b.
c.
d.
AB CD and AD BC
AB || CD and AD || BC
AB CD and AB || CD
AC BD and AB || CD
e.
AE  ½ AD and BE 
f.
AB BD CD AC
g.
mD mB and mA mC
A
B
E
C
D
Math 1312 Popper 11
Use for Popper 11 question 1 and question 2:
E  4x + 2°
x + 8° 


H Popper 11 question 1: Find
A.
°
B.
°
Popper 11 question 2: Find
A.
°
B.
°
∠
C.
∠
C.
F G in parallelogram EFGH.
°
D.
°
D. None of these
in parallelogram EFGH.
°
D.
°
D. None of these
Popper 11 question 3: Which of the following is NOT a
quadrilateral?
A. rhombus
B. parallelogram
C. triangle
D. square
Popper 11 question 4: Which quadrilateral has four (4) congruent
sides?
A. rhombus
B. trapezoid
C. rectangle
D. parallelogram
Popper 11 question 5: Which of the following has only one (1) pair of parallel
sides?
A. square
B. rectangle
C. parallelogram
D. trapezoid
Math 1312
Popper 09
Popper 9 question 1: In ∆DEF, If DE = DF and mD =100˚, then ∆DEF is
Draw a picture!!
A. acute and isosceles
C. right and isosceles
B. right and scalene
D. obtuse and isosceles
Use for Popper 9 questions 2 and 3
Given isosceles ISO with base SO . Find the measure of each angle.
S O mS = 5x – 18
mO = 2x + 21.
I Popper 9 question 2: Find the measure of angle S.
A.
°
B.
°
C.
°
D.
°
E. None of these
Popper 9 question 3: Find the measure of angle I.
A.
°
B.
°
C.
°
D.
Popper 09 question 4:
A. ∠
B. ∠
C. ∠
°
E. None of these
Math 1312 Popper 09
Popper 09 question 5:
A. SAS
B. AAS
C. ASA
D. SSS
Math 1312
Popper 10
1.
A. 80
B. 50
C. 130
D. 105
2.
A. 111
B. 63
C. 42
D. 84
Math 1312
Popper 10
3.
A. 15
B. 80
C. 30
D. 60
4. Two of the sides of an isosceles triangle have lengths of 16 cm and 7 cm. What is
the length of the base?
A. 10 cm
B. 7 cm
C. 16 cm D. 14 cm
5. Determine whether it is possible to draw a triangle with sides of the given measures. 1. 4, 2, 1 2. 3, 3, 3 3. 10, 5, 10 A. 1 only B. 1 and 2 only C. 2 and 3 only D. 3 only E. None of these