QuadrilateralsQuadrilateralsQuadrilateralsQuadrilaterals

§§ 8.1 Quadrilaterals 8.1 Quadrilaterals

§§ 8.4 Rectangles, Rhombi, and Squares 8.4 Rectangles, Rhombi, and Squares

§§ 8.3 Tests for Parallelograms 8.3 Tests for Parallelograms

§§ 8.2 Parallelograms 8.2 Parallelograms

§§ 8.5 Trapezoids 8.5 Trapezoids

QuadrilateralsQuadrilaterals

You will learn to identify parts of quadrilaterals and find thesum of the measures of the interior angles of a quadrilateral.

1) Quadrilateral2) Consecutive3) Nonconsecutive4) Diagonal

QuadrilateralsQuadrilaterals

A quadrilateral is a closed geometric figure with ____ sides and ____ vertices.four four

The segments of a quadrilateral intersect only at their endpoints.

Quadrilaterals Not Quadrilaterals

Special types of quadrilaterals include squares and rectangles.

QuadrilateralsQuadrilaterals

A

B

CD

Quadrilaterals are named by listing their vertices in order.

There are several names for the quadrilateral below.

Some examples:

quadrilateral ABCD

quadrilateral BCDA

quadrilateral CDAB or

quadrilateral DABC

S R

Q

P

QuadrilateralsQuadrilaterals

Any two _______ of a quadrilateral are either __________ or _____________.

consecutivenonconsecutive

sidesverticesangles

QuadrilateralsQuadrilaterals

S R

Q

P

Segments that join nonconsecutive vertices of a quadrilateral are called________.diagonals

S and Q arenonconsecutivevertices. diagonal a is SQ

R and P arenonconsecutivevertices. diagonal a is RP

QuadrilateralsQuadrilaterals

Q

T

S

R

Name all pairs of consecutive sides:

Name all pairs of nonconsecutive angles:

Name the diagonals:

RS and QR ST and RS

TQ and ST QR and TQ

S and Q R and T

and QS TR

QuadrilateralsQuadrilaterals

D

C

B

AConsidering the quadrilateral to the right.

What shapes are formed if a diagonal is drawn? ___________two triangles

1

23

45

6

Use the Angle Sum Theorem (Section 5-2)to find m1 + m2 + m3 180

Use the Angle Sum Theorem (Section 5-2)to find m4 + m5 + m6 180

Find m1 + m2 + m3 + m4 + m5 + m6

180+ 180

360

This leads to the following theorem.

QuadrilateralsQuadrilaterals

Theorem

8-1

The sum of the measures of the angles of a quadrilateral is

____.360

a°

d°

c°

b°

a + b + c + d = 360

QuadrilateralsQuadrilaterals

A

D

C

BmA + mB + mC + mD = 360

x + 2x + x – 10 + 50 = 360

Find the measure of B in quadrilateral ABCD if A = x, B = 2x,C = x – 10, and D = 50.

4x + 40 = 360

4x = 320

x = 80

B = 2x

B = 2(80)

B = 160

QuadrilateralsQuadrilaterals

ParallelogramsParallelograms

You will learn to identify and use the properties of parallelograms.

1) Parallelogram

ParallelogramsParallelograms

A parallelogram is a quadrilateral with two pairs of ____________.parallel sides

A B

D C

In parallelogram ABCD below, and CBDA || DCAB ||

Also, the parallel sides are _________.congruent

Knowledge gained about “parallels” (chapter 4)will now be used in the following theorems.

Theorem8-2

Theorem8-3

Theorem8-4

ParallelogramsParallelograms

Opposite angles of a parallelogram are ________.

Opposite sides of a parallelogram are ________.

The consecutive angles of a parallelogram are ____________.

A B

D C

A B

D C

A B

D C

A C and B D

DCABCBDA || and ,||

mA + mB = 180mD + mC = 180

congruent

congruent

supplementary

ParallelogramsParallelograms

In RSTU, RS = 45, ST = 70, and U = 68.

R

U

S

T

45

70

68°

Find:

RU = ____

UT = _____

mS = _____

mT = _____

70 Theorem 8-3

45 Theorem 8-3

68° Theorem 8-2

112° Theorem 8-4

ParallelogramsParallelograms

Theorem

8-5

The diagonals of a parallelogram ______ each other.

A

D

B

C

E

bisect

ECAE

EBDE

In RSTU, if RT = 56, find RE. R

U

S

T

E

RE = 28

RTRE2

1

56RE2

1

A

D

B

C

ParallelogramsParallelograms

In the figure below, ABCD is a parallelogram.

DB BD

Since AD || BC and diagonal DB is a transversal, then ADB CBD.

(Alternate Interior angles)

Since AB || DC and diagonal DB is a transversal, then BDC DBA.

(Alternate Interior angles)

BDCDBA

ASA Theorem

ParallelogramsParallelograms

Theorem

8-6

A diagonal of a parallelogram separates it into two_________________.

A

D

B

C

congruent triangles

BDCDBA

ParallelogramsParallelograms

The Escher design below is based ona _____________.

You can use a parallelogram to make a simpleEscher-like drawing.

Change one side of the parallelogram and thentranslate (slide) the change to the opposite side.

The resulting figure is used to make a designwith different colors and textures.

parallelogram

ParallelogramsParallelograms

Tests for ParallelogramsTests for Parallelograms

You will learn to identify and use tests to show that a quadrilateral is a parallelogram.

Nothing New!

Tests for ParallelogramsTests for Parallelograms

Theorem

8-7

If both pairs of opposite sides of a quadrilateral are _________, then the quadrilateral is a parallelogram.

A

D C

B

congruent

BCAD

DCAB

and

Tests for ParallelogramsTests for Parallelograms

You can use the properties of congruent triangles and Theorem 8-7 to findother ways to show that a quadrilateral is a parallelogram.

In quadrilateral PQRS, PR and QS bisect eachother at T.

Show that PQRS is a parallelogram by providing a reason for each step.

TSQTTRPT and Definition of segment bisector

QTRSTPRTSPTQ and Vertical angles are congruent

RTQPTSRSTPQT and

RQPSRSPQ and

ramparallelog a is PQRS

SAS

Corresp. parts of Congruent Triangles are Congruent

Theorem 8-7

T

P

S R

Q

Tests for ParallelogramsTests for Parallelograms

Theorem

8-8

If one pair of opposite sides of a quadrilateral is _______ and _________, then the quadrilateral is a parallelogram.

A

D C

B

congruent

DCAB DCAB ||

parallel

Tests for ParallelogramsTests for Parallelograms

Theorem

8-9

If the diagonals of a quadrilateral ________________,then the quadrilateral is a parallelogram.

EBDE ECAE

bisect each other

A

D C

B

E

Tests for ParallelogramsTests for Parallelograms

Determine whether each quadrilateral is a parallelogram.If the figure is a parallelogram, give a reason for your answer.

A

D C

B

DCAB ||

DCAB Given

Alt. Int. Angles

Therefore, quadrilateral ABCD is a parallelogram. Theorem 8-8

Tests for ParallelogramsTests for Parallelograms

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

You will learn to identify and use the properties of rectangles,rhombi, and squares.

1) Rectangle2) Rhombus3) Square

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

A closed figure,4 sides & 4 verticesQuadrilateral

Opposite sides parallelopposite sides congruent

Parallelogram

Parallelogram with4 right angles

RectangleParallelogram with4 congruent sidesRhombus

Parallelogram with

4 congruent sides and

4 right angles

Square

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

Identify the parallelogram below.

D C

BA

Parallelogram ABCD has4 right angles, but the foursides are not congruent.

Therefore, it is a _________rectangle

Identify the parallelogram below.

rhombus

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

Theorem

8-10

The diagonals of a rectangle are _________.congruent

A

D

B

C

DBAC

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

Theorem

8-11

The diagonals of a rhombus are ____________.perpendicular

A

B

D

C

DBAC

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

Theorem

8-12

Each diagonal of a rhombus _______ a pair of opposite angles.bisects

43

8

7

6

5

43

21

21

65

87 D

C

B

A

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

Use square XYZW to answer the following questions:

W Z

YX

O

1) If YW = 14, XZ = ____

2) mYOX = ____

A square has all the properties of a rectangle, and the diagonals of a rectangle are congruent.

14

A square has all the properties of a rhombus, and the diagonals of a rhombus are perpendicular.

90

3) Name all segments that are congruent to WO. Explain your reasoning.

The diagonals are congruent and they bisecteach other.

OY, XO, and OZ

Quadrilaterals

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

Parallelograms

Rhombi RectanglesSquares

Use the Venn diagram to answer the following questions: T or F

1) Every square is a rhombus: ___

2) Every rhombus is a square: ___

3) Every rectangle is a square: ___

4) Every square is a rectangle: ___

5) All rhombi are parallelograms: ___

6) Every parallelogram is a rectangle: ___

T

T

TF

F

F

Rectangles, Rhombi, and SquaresRectangles, Rhombi, and Squares

TrapezoidsTrapezoids

You will learn to identify and use the properties of trapezoidsand isosceles trapezoids.

1) Trapezoid

Trapezoids Trapezoids

A trapezoid is a ____________ with exactly one pair of ____________.quadrilateral parallel sides

T

P A

R

The parallel sides are called ______.bases

base

baseThe non parallel sides are called _____.legs

leg leg

Each trapezoid has two pair ofbase angles.

base anglesT and R are one pairof base angles.

P and A are the other pair of base angles.

Trapezoids Trapezoids

Theorem

8-13

The median of a trapezoid is parallel to the _____, basesand the length of the median equals _______________ of the lengths of the bases.

one-half the sum

C

N

B

D

M

A

MNDCMNAB || ,||

DCABMN 2

1

Trapezoids Trapezoids

C

N

D

M

BA

Find the length of median MN in trapezoid ABCD if AB = 16 and DC = 20

16

20

DCABMN 2

1

20162

1MN

362

1MN

18MN

18

Trapezoids Trapezoids

If the legs of a trapezoid are congruent, the trapezoid is an _________________.isosceles trapezoid

In lesson 6 – 4, you learned that the base angles of an isosceles triangle arecongruent.

There is a similar property of isosceles trapezoids.

Trapezoids Trapezoids

Theorem

8-14

Each pair of __________ in an isosceles trapezoid is congruent.base angles

Z Y

XW

XW

YZ

Trapezoids Trapezoids

T

P A

R

60°

Find the missing angle measures in isosceles trapezoid TRAP.

P A

mP = mA

60 = mA

60°

Theorem 8 – 14

T R

P + A + 2(T) = 360

Theorem 8 – 14

60 + 60 + 2(T) = 360

120 + 2(T) = 360

2(T) = 240

T = 120

120°

R = 120

120°

Trapezoids Trapezoids