objective: prove quadrilateral conjectures by using triangle congruence postulates and theorems

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 SquareAlike:

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 supplementaryDiagonals bisect each otherA 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 DiagonalsPerpendicular Sides

PROPERTIES OF SPECIAL QUADRILATERALS:

RHOMBUSES:Rhombuses have all of the properties of parallelograms plus:

Four congruent sides

Perpendicular diagonalsDiagonals 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 parallelogramQ R

ST

QR = ______6x-2 10

x+4

Example:Find the indicated measure for the parallelogramC

F E

DCD = ______

(𝟐𝒙+𝟔)𝟎(

x-7

Example:Find the indicated measure for the parallelogramM

P O

Nm<N = ______

(𝒙−𝟒)𝟎

(

Example:Find the indicated measure for the parallelogram E

G

FH 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

CA AD = ______(

(𝟒 𝒙−𝟕

Example:Find the indicated measure for the parallelogramD

G F

Em<E = ______

(

(

Example:Find the indicated measure for the parallelogramQ R

ST

QR = ______

−𝒙+𝟐𝟒(

(

Example:Find the indicated measure for the parallelogramP

S R

Qm<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.