Objective:
Prove quadrilateral conjectures by using triangle
congruence postulates and theorems
Warm-Up:
How are the quadrilaterals in each pair alike?
How are they different?
Parallelogram vs Square
Alike: Opp sides || & ≅
Rhombus vs Square
Alike: 4 = sides
Opp <‘s =
Diagonals perp.
Different: Sq 4 right <‘s
Sq 4 ≅ sides Different: Sq has 4 right <‘s
Quadrilateral: Any four sided polygon.
Trapezoid: A quadrilateral with one and only one
pair of parallel sides.
Parallelogram: A quadrilateral with two pairs of
parallel sides.
Rhombus:A quadrilateral with four congruent sides.
Rectangle: A quadrilateral with four right angles.
Square: A quadrilateral with four congruent sides
and four right angles.
PROPERTIES OF SPECIAL QUADRILATERALS:
PARALLELOGRAMS:
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
Both pairs of opposite sides angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
A diagonal creates two congruent triangles
(it’s a turn – NOT a flip)
P
M
L
G
Theorem:
A diagonal of a parallelogram divides the parallelogram into two congruent triangles.
PROPERTIES OF SPECIAL QUADRILATERALS:
RECTANGLES:
Rectangles have all of the properties of
parallelograms plus:
Four right angles
Congruent Diagonals
Perpendicular Sides
PROPERTIES OF SPECIAL QUADRILATERALS:
RHOMBUSES:
Rhombuses have all of the properties of
parallelograms plus:
Four congruent sides
Perpendicular diagonals
Diagonals bisect each other
PROPERTIES OF SPECIAL QUADRILATERALS:
SQUARES:
Squares have all of the properties of
parallelograms, rectangles & rhombuses.
Parallelogram
Rhombus Square Rectangle
Note: Sum of the interior <‘s of a quadrilateral = _____
Example:
Find the indicated measures for the parallelogram WXYZ
W
5
X
2.2
Z
𝟐𝟓𝟎
𝟏𝟐𝟎𝟎
Y
m<WXZ = _____
m<ZXY = _____
m<W = _____
XY = _____
m<WZX = _____
Perimeter of
WXYZ= _____
Example: ABDE is a parallelogram & BC ≅ BD
A
B
E
D
C
If m<BDC = 𝟔𝟐𝟎 , find m<EAB. _______
If m<DBC = 𝟑𝒙, m<BCD=6x, find m<EAB ______
If m<DBC = 𝟑𝒙, m<BCD=6x, find m<ABD ______
Example:
Find the indicated measure for the parallelogram
A
𝟑
𝟐
( 𝒙 +𝟑𝟎)𝟎
D
(𝟐𝒙)𝟎
C
B
m<A = ______
Example:
Find the indicated measure for the parallelogram
R
Q
QR = ______
6x-2
T
x+4
10
S
Example:
Find the indicated measure for the parallelogram
C
D
(𝟓𝟎)𝟎
(𝟐𝒙 + 𝟔)𝟎
F
x-7
E
CD = ______
Example:
Find the indicated measure for the parallelogram
M
N
(𝟔𝒙 + 𝟏𝟔)𝟎
P
(𝒙 − 𝟒)𝟎
m<N = ______
O
Example:
Find the indicated measure for the parallelogram
E
H
m<G = ______
(𝟐𝟕)𝟎
F
G
Homework: Practice
Worksheet
Objective:
Identify the missing component of a given
parallelogram through the use of factoring.
Warm-Up:
What is the first number that
has the letter “a” in its name?
Example:
Find the indicated measure for the parallelogram
B
A
AD = ______
(𝒙𝟐 − 𝟑𝒙 − 𝟒)𝟎
(𝟐𝒙 + 𝟐𝟎)𝟎
𝟒𝒙 − 𝟕
D
C
Example:
Find the indicated measure for the parallelogram
D
(−𝟔𝒙 +
G
(𝟐𝒙𝟐 + 𝟒𝒙 − 𝟖)𝟎
F
𝟔𝟒)𝟎
E
m<E = ______
Example:
Find the indicated measure for the parallelogram
R
Q
T
−𝒙 + 𝟐𝟒
QR = ______
S
Example:
Find the indicated measure for the parallelogram
P
(−𝟑𝒙 +
S
(𝟐𝒙𝟐 + 𝟑𝒙 − 𝟑𝟖)𝟎
R
𝟕𝟎)𝟎
Q
m<R = ______
Collins Writing:
How could you determine the sum of the
interior angles of a quadrilateral?
Homework: Practice
Worksheet
Given: Parallelogram PLGM with diagonal LM
Prove: ∆LGM ≅ ∆MPL
STATEMENTS
REASONS
M
P
4
3
2
G
L
1
Given: Parallelogram ABCD with diagonal BD
Prove: ∆ABD ≅ ∆CDB
STATEMENTS
REASONS
D
A
1
3
2
5
6
C
B
4
Theorem: Opposite sides of a parallelogram are congruent.
Given: Parallelogram ABCD with diagonal BD
Prove: AB ≅ CD & AD ≅ CB
STATEMENTS
REASONS
Theorem: Opposite angles of a parallelogram are congruent.
Given: Parallelogram ABCD with diagonals BD & AC
Prove: <BAD ≅ <DCB & <ABC ≅ <CDA
STATEMENTS
REASONS