Objective:Prove quadrilateral conjectures by using triangle congruence postulates and theorems

4.5 Properties of Quadrilaterals

Warm-Up:How are the quadrilaterals in each pair alike? How are they different?

Parallelogram vs Square

Rhombus vs Square

Alike:

Different:

Alike:

Different:

Opp sides || & 4 = sidesOpp <‘s = Diagonals perp.

Sq has 4 right <‘s

Sq 4 right <‘sSq 4 sides

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.Rhombu

s: 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 parallelBoth pairs of opposite sides are congruentBoth pairs of opposite sides angles are congruentConsecutive angles are supplementary

Diagonals bisect each other

A diagonal creates two congruent triangles (it’s a turn – NOT a flip)

M

LP

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 Rectangle

Square

Note: Sum of the interior <‘s of a quadrilateral = _____

Example:

Find the indicated measures for the parallelogram WXYZ

m<WXZ = _____

m<W = _____

m<ZXY = _____

XY = _____

m<WZX = _____Perimeter of WXYZ= _____

W X

Z Y

2.2

5

𝟐𝟓𝟎 𝟏𝟐𝟎𝟎

Example: ABDE is a parallelogram & BC BD

If m<BDC = , find m<EAB. _______

A B

DE C

If m<DBC = , m<BCD=6x, find m<EAB ______

If m<DBC = , m<BCD=6x, find m<ABD ______

Example:Find the indicated measure for the parallelogramA

B

C

D

m<A = ______

(𝟐 𝒙)𝟎

(

Example:Find the indicated measure for the parallelogram

QR

ST

QR = ______6x-2 10

x+4

Example:Find the indicated measure for the parallelogramC

F E

D

CD = ______(𝟐 𝒙+𝟔)𝟎

(

x-7

Example:Find the indicated measure for the parallelogram

M

P O

N

m<N = ______

(𝒙−𝟒)𝟎

(

Example:

Find the indicated measure for the parallelogram

E

G

F

H m<G = ______(

Homework:Practice Worksheet

Objective:Identify the missing component of a given parallelogram through the use of factoring.

Parallelograms & 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

D

C

A AD = ______(

(𝟒 𝒙−𝟕

Example:Find the indicated measure for the parallelogram

D

G F

E

m<E = ______

(

(

Example:Find the indicated measure for the parallelogram

QR

ST

QR = ______

−𝒙+𝟐𝟒(

(

Example:Find the indicated measure for the parallelogram

P

S R

Q

m<R = ______

(

(

Collins Writing:

How could you determine the sum of the interior angles of a quadrilateral?

Homework:Practice Worksheet

L

G

P

M4

2

3

1Given: Prove:

Parallelogram PLGM with diagonal LM∆LGM ∆MPL

STATEMENTS REASONS

Given: Prove:

Parallelogram ABCD with diagonal BD∆ABD ∆CDB

STATEMENTS REASONS C

A

D2

1

5

4

B3

6

Given: Prove:

Parallelogram ABCD with diagonal BDAB CD & AD CB

STATEMENTS REASONS

Theorem: Opposite sides of a parallelogram are congruent.

Given: Prove:

Parallelogram ABCD with diagonals BD & AC<BAD <DCB & <ABC <CDA

STATEMENTS REASONS

Theorem: Opposite angles of a parallelogram are congruent.