Chapter 6. Formed by 3 or more segments (sides) Each side intersects only 2 other sides (one at each endpoint)

Chapter 6

Quadrilaterals

Polygons

What is a Polygon?Formed by 3 or more segments (sides)

Each side intersects only 2 other sides (one at each endpoint)

What is a Polygon?

Number of Sides

Name of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n-gon

Polygons

are

named by

the

number

of sides

they have

What’s in a name

?

CONVEX

Classifying PolygonsCONCAVE

Concave or Convex?

Regular Polygons:Equilateral & Equiangular

Classifying Polygons

Regular or Irregular?

Segment that joins 2 non-consecutive vertices.

Diagonals of Polygons

Diagonals

Interior Angles of a Quadrilateral Theorem

The Sum of the Measures of the Interior Angles of a Quadrilateral is 360°

Interior Angles of Quadrilaterals

Solve for x…

Parallelograms

QuadrilateralBoth pairs of opposite sides are

parallel

What is a Parallelogram?

OPPOSITE SIDES are congruent

If a Quadrilateral is a Parallelogram, Then….

OPPOSITE ANGLES are congruent

Theorems about Parallelograms

CONSECUTIVE ANGLES are supplementary

If a Quadrilateral is a Parallelogram, Then….

DIAGONALS bisect each other

Theorems about Parallelograms

A + B = 180°

Proving Quadrilaterals are Parallelograms

If both pairs of opposite sides of a quad. are …

If both pairs of opposite angles of a quad. are …

If an angle of a quad. is supplementary to both of its consecutive angles …

If the diagonals of a quad. bisect each other…

Then, the Quadrilateral

is a Parallelogram.

Prove it!Proving Quadrilaterals are Parallelograms…

If one pair of opposite sides of a quadrilateral are congruent AND parallel

Then, the Quadrilateral

is a Parallelogram.

Prove it!Proving Quadrilaterals are Parallelograms…

Describe how to prove that ABCD is a parallelogram given that ∆PBQ ∆RDS and ∆PAS ∆RCQ.

Prove it!Let’s practice….

Let’s practice….Prove that EFGH is a

parallelogram by showing that a pair of opposite sides are both congruent and parallel.

Use E(1, 2), F(7, 9), G(9, 8), and H(3, 1).

Prove it!Prove that JKLM is a

parallelogram by showing that the diagonals bisect each other.

Use J(-4, 4), K(-1, 5), L(1, -1), and M(-2, -2).

Quiz 1Sections 1, 2, & 3

Special Parallelograms

RhombusA parallelogram with 4 congruent sides

Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides.

Theorem 6.11:A parallelogram is a rhombus if

and only if its diagonals are perpendicular.

ABCD is a rhombus if and only if AC BD

Rhombus

Theorem 6.12:A parallelogram is a rhombus if

and only if its diagonals bisect a pair of opposite angles.

ABCD is a rhombus if and only if AD bisects CAB and BDC and BC bisects DCA and ABD

Rhombus

RectangleA parallelogram with 4 right angles

Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.

Theorem 6.13:A parallelogram is a rectangle if

and only if its diagonals are congruent.

ABCD is a rectangle if and only if AC BD

Rectangle

SquareA parallelogram with 4 congruent sides AND 4 right angles

Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Special Parallelograms

Trapezoids

TrapezoidsQuadrilateral with

only one pair of parallel sides.Parallel sides are

the “bases”Non-parallel sides

are the “legs”Has 2 pairs of base

angles

Base Angles

Isosceles TrapezoidsShow that RSTV is a

trapezoid…

Isosceles TrapezoidsLegs are congruent

If mA = 45°, What is the measure of B?

What is the measure of C?

What is the measure of D?

Isosceles TrapezoidsTheorem 6.14:

If a trapezoid is isosceles, then each pair of base angles is congruent

A D, B C

Isosceles TrapezoidsTheorem 6.15: (Converse to theorem 6.14)

If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid

ABCD is an isosceles trapezoid

Isosceles TrapezoidsTheorem 6.16:

A trapezoid is isosceles if and only if its diagonals are congruent

ABCD is isosceles if and only if AC BD

Midsegment Theorem for Trapezoids

(Theorem 6.17)

EF AB, EF DC, EF = ½(AB + DC)

The midsegment of a trapezoid is …Parallel to each

base½ the sum of the

length of the bases

Trapezoids

Kites

KitesA quadrilateral that

has two pairs of consecutive congruent sides. Opposite sides are

NOT congruent.

Theorem 6.18: If a quadrilateral is a kite, then its diagonals are perpendicular

Theorems about

Kites

KT EI

If KS = ST = 5, ES = 4, and KI = 9, What is the measure of EK?What is the measure of SI?

Practicing Theorems about

Kites

Theorem 6.19: If a quadrilateral is a kite, then only one pair of opposite angles are congruent

Theorems about

Kites

K M, J L

If mJ = 70 and mL = 50, What is mM & mK?

Practicing Theorems about

Kites

Quiz 2Sections 4 & 5

Special Quadrilaterals

When you join the midpoints of the sides of ANY quadrilateral, what special quadrilateral is formed? Explain.

On a piece of graph paper… Draw ANY quadrilateralFind and connect the midpoints of each

sideWhat type of Quadrilateral is formed?How do you know?

Special Quadrilaterals

Let’s prove a quadrilateral is a “special” shape…Use the Definition of the ShapeUse a Theorem

EXAMPE: Show that PQRS is a rhombusHow would you

prove this to be true?

Special Quadrilaterals

Create a Graphic Organizer showing the relationship between the following

figures…

Isosceles Trapezoid

KiteParallelogramQuadrilaterals

RectangleRhombusSquareTrapezoid

Special Quadrilaterals

Requirements..•Accurate Graphic Organizers

•Each figure should include an picture and description

• Bold, Clear, and Colorful

Graphic Organizer ExamplesSpecial Quadrilaterals

Areas

Area of a RectangleA = bh Find the area of the

rectangle below:

Area of a ParallelogramA = bh Find the area of the

parallelogram below:

Area of a TriangleA = ½bh Find the area of the

triangles below:

Area of a Triangle (again)A = ½bh What is the height

the triangle below:

A = 27 ft2

Base = 9 feet

Area of a TrapezoidA = ½h(b1 + b2)

Find the area of the trapezoid below:

Area of a KiteA = ½d1d2

Find the area of the kite below:

Area of a RhombusA = ½d1d2

Find the area of the rhombus below:

Quiz 3Sections 6 & 7