Quadrilaterals and Polygons Polygon: A plane figure that is formed by three or more segments (no two of which are collinear), and each segment (side) intersects.

Quadrilaterals and Polygons

Polygon: A plane figure that is formed by three or more segments (no two of which are collinear), and each segment (side) intersects at exactly two other sides – one at each endpoint (Vertex).

Which of the following diagrams are polygons?

Polygons are Named & Classified by the Number of Sides They Have

# of Sides Type of Polygon

3

4

5

6

7

# of Sides Type of Polygon

8

9

10

12

#

What type of polygons are the following?

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodagon

N-gon

Convex and Concave Polygons

Interior

Convex – A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon.Concave – A polygon that is not convex

Interior

Equilateral, Equiangular, and Regular

Diagonals and Interior Angles of a Quadrilateral

Diagonal – a segment that connects to non-consecutive vertices.

Theorem 6.1 – Interior Angles of a Quadrilateral TheoremThe sum of the measures of the interior angles of a quadrilateral is 360O

m<1 + m<2 + m<3 + m<4 = 360o

80o 70o

xo 2xo

Properties of Parallelograms

Theorem 6.2If a quadrilateral is a parallelogram, then its opposite sides are congruent. PQ = RS and SP = QR

Theorem 6.3If a quadrilateral is a parallelogram, then it opposite angles are congruent. <P = < R and < Q = < S

Theorem 6.4If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. m<P + m<Q = 180o, m<Q + m<R = 180o

m<R + m<S = 180o, m<S + m<P = 180o

Theorem 6.5If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM = SM and PM = RM

__ __~~

~~

_~~

___

Q R

P S

Q R

P S

Q R

P S

Q R

P S

Using the Properties of Parallelograms

FGHJ is a parallelogram.Find the length of:

a. JHb. JK

F 5 G

K 3

J H

PQRS is a parallelogram.Find the angle measures:

a. m<Rb. m<Q

Q R

70o

P S

PQRS is a parallelogram.Find the value of x

P Q

3xo 120o

S R

Proofs Involving Parallelograms A E B 2

1D C

3G F

Given: ABCD and AEFG are parallelograms

Prove: <1 = < 3

Statements Reasons

~

Plan: Show that both angles are congruent to <2

1. ABCD & AEFG are Parallelograms 1. Given2. <1 = < 2 2. Opposite Angles are congruent (6.3)3. <2 = <3 3. Opposite Angles are Congruent (6.3)4. <1 = <3 4. Transitive Property of Congruence~

~~

Proving Theorem 6.2

Given: ABCD is a parallelogram

Prove: AB = CD, AD = CB

Statements Reasons

~~____

A B

D C

Plan: Insert a diagonal which will allow us to divide the parallelogram into two triangles

1. ABCD is a parallelogram 1. Given2. Draw Diagonal BD 2. Through any two points there

exists exactly one line3. AB || CD, and AD || CB 3. Def. of a parallelogram

________

__

4. <ABD = < CDB 4. Alternate Interior Angles Theorem5. <ADB = < CBD 5. Alternate Interior Angles Theorem6. DB = DB 6. Reflexive Property of Congruence7. /\ ADB = /\ CBD 7. ASA Congruence Postulate8. AB = CD, AD = CB 8. CPCTC

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Proving Quadrilaterals are Parallelograms

Theorem 6.6If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Q R

P S

Q R

P S

Theorem 6.9If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Q R

P S

(180-x)o xo

xo

Theorem 6.8If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram

Q R

P S

Theorem 6.7If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Concept Summary – Proving Quadrilaterals are Parallelograms

• Show that both pairs of opposite sides are

• Show that both pairs of opposite sides are

•Show that both pairs of opposite angles are

• Show that one angle is supplementary to

• Show that the diagonals

• Show that one pair of opposite sides are both

congruent

parallel

congruent

BOTH consecutive interior <‘s

bisect each other

congruent and ||

Proving Quadrilaterals are Parallelograms – Coordinate Geometry

C(6,5)

B(1,3)

D (7,1)

A(2, -1)

How can we prove that the Quad is a parallelogram?

1. Slope - Opposite Sides ||

2. Length (Distance Formula) – Opposite sides same length

3. Combination – Show One pair of opposite sides both || and congruent

Rhombuses, Rectangles, and Squares

Rhombus – a parallelogram with four congruent sides

Rectangle – a parallelogram with four right angles

Square – a parallelogram with four congruent sides and four right angles

Parallelograms

RhombusesSquares

Rectangles

Using Properties of Special Triangles

If ABCD is a rectangle, what else do you know about ABCD?

A B

C D

Corollaries about Special Quadrilaterals

Rhombus Corollary – A quad is a rhombus if and only if it has four congruent sidesRectangle Corollary – A quad is a rectangle if and only if it has four right anglesSquare Corollary – A quad is a square if and only if it is a rhombus and a rectangle

How can we use these special properties and corollaries of a Rhombus? P Q

S R

2y + 3

5y - 6

Using Diagonals of Special Parallelograms

Theorem 6.11: A parallelogram is a rhombus if and only if its diagonals are perpendicular

Theorem 6.12: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

Theorem 6.13: A parallelogram is a rectangle if and only if its diagonals are congruent

Decide if the statement is sometimes, always, or never true.1. A rhombus is equilateral.

2. The diagonals of a rectangle are _|_.

3. The opposite angles of a rhombus are supplementary.

4. A square is a rectangle.

5. The diagonals of a rectangle bisect each other.

6. The consecutive angles of a square are supplementary.

Always

Sometimes

Sometimes

Always

Always

Always

Quadrilateral ABCD is Rhombus.7. If m <BAE = 32o, find m<ECD.8. If m<EDC = 43o, find m<CBA.9. If m<EAB = 57o, find m<ADC.10. If m<BEC = (3x -15)o, solve for x.11. If m<ADE = ((5x – 8)o and m<CBE = (3x +24)o, solve for x12. If m<BAD = (4x + 14)o and m<ABC = (2x + 10)o, solve for x.

A B

E

D C

32o

86o

66o

35o

16 26

Coordinate Proofs Using the Properties of Rhombuses, Rectangles and Squares

Using the distance formula and slope, how can we determine the specific shape of a parallelogram?

Rhombus –

Rectangle –

Square -

Based on the following Coordinate values, determine if each parallelogramis a rhombus, a rectangle, or square.

P (-2, 3) P(-4, 0)Q(-2, -4) Q(3, 7)R(2, -4) R(6, 4)S(2, 3) S(-1, -3)

Given: HIJK is a parallelogram/\ HOI = /\ JOI

Prove: HIJK is a Rhombus

Statements Reasons

H I

O

K J

~

Given: RECT is a Rectangle

Prove: /\ ART = /\ ACE

Statements Reasons

~

R E

A

T C

Given: PQRT is a Rhombus

Prove: PR bisects <TPQ and < QRT, and QT bisects <PTR and <PQP

Statements Reasons

P Q

T R

Plan: First prove that Triangle PRQ is congruent to Triangle PRT; and Triangle TPQ is congruent to Triangle TRQ

Trapezoids and Kites

A B

D C>

>

A Trapezoid is a Quad with exactly one pair of parallel sides. The parallel sides are the BASES. A Trapezoid has exactly two pairs of BASE ANGLES

In trapezoid ABCD, Which 2 sides are the bases? The legs? Name the pairs of base angles.

A B

D C>

>If the legs of the trapezoid are congruent, then the trapezoid is an Isosceles Trapezoid.

Theorems of Trapezoids

Theorem 6.14If a trapezoid is isosceles, then each pair of base angles is congruent.

<A = <B = <C = <D~ ~ ~

A B

D C>

>

Theorem 6.15If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

ABCD is an isosceles trapezoid

A B

D C>

>

~

Theorem 6.16A trapezoid is isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC = BD__

A B

D C>

>

Kites and Theorems about Kites

A kite is a quadrilateral that has two pairs of consecutive congruent sides,But opposite sides are NOT congruent.

Theorem 6.18If a Quad is a Kite, then its diagonals are perpendicular.

Theorem 6.19If a Quad is a kite then exactly one pair of opposite angles are congruent

Using the Properties of a Kite

X

12

20 U 12 W Y

12

Z

Find the length of WX, XY, YZ, and WZ.

J

H 132o 60o K

G

Find the angle measures of <HJK and < HGK

Summarizing the Properties of Quadrilaterals

Quadrilaterals

______________ _________________ ________________

____________ _____________ ____________ ______________

Kites Parallelograms Trapezoids

Rhombus Squares Rectangles Isosceles Trap.

Properties of Quadrilaterals

X X X X

X

X X X X X

X X X X

X X X X

X X

X X X X

X X X

Property Rectangle Rhombus Square Kite Trapezoid

Both pairs of Opp. sides a ||Exactly one pair of Opp. Sides are ||Diagonals are _|_Diagonals are =Diagonals Bisect each otherBoth pairs of Opp. Sides are =Exactly one pair of opp. Sides are =All Sides are =Both pairs of Opp. <'s are =Exactly one pair of Opp <'s are =All <'s are =

Using Area Formulas

Area of a Square PostulateThe area of a square is the square of the length of its side.

Area Congruence PostulateIf two polygons are congruent then they have the same area.

Area Addition PostulateThe area of a region is the sum of the area of its non-overlapping sides.

Area of a RectangleThe area of a rectangle is the product of its base and height.

A = bh

Area of a ParallelogramThe area of a parallelogram is the product of a base, and it’s corresponding height

A = bh

Area of a TriangleThe area of a triangle is one half the product of a base and its corresponding height

A = ½bh

h

b

h

b

h

b

Given: /\ RQP = /\ ONP R is the midpoint of MQProve: MRON is a parallelogram

Statements Reasons

~__

Q

R P O

M N

1. /\ RQP = /\ ONP 1. Given R is the midpoint of MQ

2. MR = RQ 2. Definition of a midpoint

3. RQ = NO 3. CPCTC

4. MR = NO 4. Transitive Property of Congruency

5. <QRP = < NOP 5. CPCTC

6. MQ || NO 6. Alternate Interior <‘s Converse

7. MRON is a parallelogram 7. Theorem 6.10

~

~

~

~

~

__ __

__ __

__ __

__ __

Given: UWXZ is a parallelogram, <1 = <8Prove: UVXY is a parallelogram

Statements Reasons

U V W

Z Y X

2 3 4 1

8 5 6 7

~

1. UWXZ is a parallelogram 1. Given

2. UW || ZX 2. Definition of a parallelogram

3. UV || YX 3. Segments of Congruent Segments

4. <Z = <W 4. Opposite <‘s of a parallelogram are =

5. <1 = <8 5. Given

6. <5 = <4 6. Third Angles Theorem

6. <4 = <7 7. Alternate Interior Angles Theorem

6. <5 = <7 8. Transitive Property of Congruence

7. UY || VX 9. Corresponding Angles Converse

8. UVXY is a parallelogram 10. Definition of a Parallelogram

~

~

~

~

~

__

____

____

__

Given: GIJL is a parallelogram

Prove: HIKL is a parallelogram

Statements Reasons

L K J

M

G H I

1. GIJL is a parallelogram 1. Given

2. GI || LJ 2. Definition of a parallelogram

3. <GIL = <JLI 3. Alternate Interior Angles Theorem

4. GJ Bisects LI 4. Diagonals of a parallelogram bisect

5. MI = ML 5. Definition of a Segment Bisector

6. <HMI = <KML 6. Vertical Angles Theorem

7. /\ HMI = /\ KML 7. ASA Congruence Postulate

8. MH = MK 8. CPCTC

9. HK and IL Bisect Each other 9. Definition of a Segment Bisector

10. HIKL is a parallelogram 10. Theorem 6.9

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____

__ __

__

~

~

~

~

~