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Properties of Quadrilaterals 3.2PowerPoint Presentation

Properties of Quadrilaterals 3.2

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Any four sided polygon is a quadrilateral.

Angles sum to be 360

We’ll study special quadrilaterals in this section:

Trapezoid

Isosceles Trapezoid

Parallelogram

Rhombus

Rectangle

Square

Kite

Opposite sides of a parallelogram are parallel

Opposite sides are congruent

Opposite angles of parallelograms are congruent.

Diagonals of a parallelogram bisect each other

Consecutive angles of a parallelogram are supplementary

Alternate interior angles are congruent

Homework

Properties of ParallelogramsHomework

Find a and b so that the quadrilateral is a parallelogram State the property.

a. mMJK

b. mJML

c. mJKL

d. mKJL

e. a f. b

100

80

80

30

21

7

Find d so that the quadrilateral is a parallelogram. State the property.

- mPLM
- b. mLMN
- c. d =

108

72

11

Find x and y so that the quadrilateral is a parallelogram State the property.

a. x b. y

y = 21

x = 12

Find x and y so that the quadrilateral is a parallelogram. State the property.

a. x b. y

x = 7

y = 4

Find the valuesso that the figure is a parallelogram State the property.

a. x b. y

c. a d. b

y = 15

a = 7

x = 25

b = 7

g. w h. z

e. x f. y

z = 4½

w = 4

y = 65

x = 8

Homework

Find x, y, w, and z so that the quadrilateral is a parallelogram. State the property .

a. mMNP

b. mNRP

c. mRNP

d. mRMN

e. mMQN

f. mMQR

g. x h. y i. w j. z

71

33

38

109

97

83

6.45

3.525

8

A rhombus is a parallelogram (this means it has ALL of the characteristics of a parallelogram)

In addition:

A rhombus has four congruent sides

The diagonals of a rhombus are perpendicular

The diagonals bisect opposite angles

Homework

Properties of a Rhombus (Rhombi)Homework characteristics of a parallelogram)

Rhombus

a. characteristics of a parallelogram)

f.

c.

d.

e.

b.

JM

m KJL

m KNL

m KJM

NM

JN

Homework

Find the indicated measure in rhombus JKLM

KM = 8 and JL = 6. State the property.

4

90°

3

5

53°

37

106°

A rectangle is a parallelogram (this means it has ALL the characteristics of a parallelogram)

IN ADDITION:

Four right angles

The diagonals of a rectangle are congruentandthey bisect each other

Homework

Properties of RectanglesHomework characteristics of a parallelogram)

Rectangles

Homework characteristics of a parallelogram)

In rectangle JKLM shown below, JL and MK are diagonals. If JL = 2x + 5 and MK = 4x – 11, what is x?x = 8

If mMNL = 140 answer the following?

d. mMJK

e. mNLK

f. mNLM

- mJNK
- b. mMNJ
- c. mLNK

g. mLJK

h. mLJM

90°

20°

140°

70°

70°

40°

20°

40°

(2z) characteristics of a parallelogram)

+ 11)

Homework

In rectangle ABCD shown below, find the value of x, y, and z.State the property.a. x b. y c. z

z = 12.5

y = 9

x = 5

Homework characteristics of a parallelogram)

WXYZ is a rectangle.

Find each measure

if m1 = 35.

State the property.

a.m1 b. m2 c. m3 d. m4

e. m5 f. m6 g. m7 h. m8

i. m9 j. m10 k. m11 l. m12

55°

55°

35°

35°

55°

35°

35°

55°

70°

110°

110°

70°

Homework characteristics of a parallelogram)

Quadrilateral JKMN is a

rectangle. Find each

measure.

State the property.

36

a. If NQ = 5x +3 & QM =4x +6, find NK.

b. If NQ =2x +3 & QK 5x -9, find JQ.

c. If NM =2x +14 & JK =x2 -1, find JK.

d. If mNJM =2x +3 & mKJM =x +6, find x.

e. If mNKM =x2 +4 & mKNM =x +30, find mJKN.

f. If mJKN =16x & mNKM = 14x, find x.

11

8 or 24

27

37

3

in. characteristics of a parallelogram)

c = 1737

Homework

Television screens are rectangles and are measured by their diagonals. Find the length of the diagonal.a² + b² = c²

21² + 36² = c²

1737 = c²

c 41.6773

A square is a parallelogram, a rectangle, and a rhombus (It has ALL those characteristics!!!)

Has four congruent sides

Has four right angles

The diagonals of a square:

bisect each other

are congruent

are perpendicular.

bisect opposite angles

Homework

Properties of SquaresHomework has ALL those characteristics!!!)

A has ALL those characteristics!!!)

B

10 in.

c = 200

C

D

Homework

Parallelogram ABCD is a square.Find x and y.a² + b² = c²

10² + 10² = c²

200 = c²

c 14.14

- x
- b. y

x = 45

y 14.14

Homework has ALL those characteristics!!!)

Inheritanceof PropertiesHomework has ALL those characteristics!!!)

2 pair of consecutive congruent sides has ALL those characteristics!!!)

Opposite sides are NOT congruent

Angles are congruent as marked

(also mK mT)

Diagonals are perpendicular

Notice only ONE diagonal

is bisected

Homework

Properties of a Kite:A quadrilateral with NO parallel sides.Homework has ALL those characteristics!!!)

Kitesx + 4 has ALL those characteristics!!!)

14

y + 16

2x + 12

Homework

Find the value of x and y. Find the lengths of the sides.- x
- b. y

10

16

c. IT

d. KE

14

32

Homework has ALL those characteristics!!!)

Find the value of x and y in the kite below.a² + b² = c²

24² + (SO)² = 27²

576 + (SO)² = 729

(SO)² = 153

SO = 153

SO 12.4

12.4

- x b. y

4x + 3 = 15

2x + 5y = 12.4

6 + 5y = 12.4

4x = 12

x = 3

5y = 6.4

y = 1.28

2 has ALL those characteristics!!!)

6

5

4

3

1

52

7

27

Homework

In kite ABCD, find the measures of the numbered angles.

a.m1 b. m2 c. m3 d. m4

e. m5 f. m6 g. m7

90°

52°

38°

27°

63°

38°

63°

Homework has ALL those characteristics!!!)

Trapezoid

Homework has ALL those characteristics!!!)

Isosceles Trapezoid

Homework has ALL those characteristics!!!)

Properties of a Trapezoid- A trapezoid has one and only one pair of parallel sides.
- The median of a trapezoid is parallel to the bases, and the length of the median equals one-half the sum of the lengths of the bases.

Base

Median

Base

6 has ALL those characteristics!!!)

65

18

Homework

For isosceles trapezoid XYZW, Find the length of the median, mX and mZ.- Median
- b. mZ
- c. mX

12

115°

65°

Homework has ALL those characteristics!!!)

In trapezoid QRST, A and B are midpoints of the legs. Find AB, mQ, and mS.a. AB b. mQ c. mS

135°

60°

16

Homework has ALL those characteristics!!!)

PQRS is an isosceles trapezoid; find x.

2a2 – 54 = a2 + 27

a2 = 81

a = 9 or a = –9

17x has ALL those characteristics!!!)

22.5x + 9

30x + 12

Homework

XY is the midsegment of trapezoid ABCD; find x.

x = 3

Homework has ALL those characteristics!!!)

1. Opposite sides parallel.

2. Opposite sides congruent.

3. Opposite angles are congruent.

4. Consecutive angles are supplementary.

5. Diagonals bisect each other.

1. Has 4 right angles.

2. Diagonals are congruent.

3. All properties of parallelogram.

1. Has 4 Congruent sides

2. Diagonals bisect opposite angles.

3. Diagonals are perpendicular.

4. All properties of parallelograms.

1. 4 congruent sides and 4 congruent

(right) angles

2. All properties of parallelogram,

rectangle, and rhombus

QUADRILATERALS

1. One pair of parallel sides

2. Leg angles supplementary

3. Midsegment = ½ (b1 + b2)

1. 2 pairs of consecutive sides congruent

2. 1 pair of opposite angles congruent

3. Diagonals perpendicular

4. Small diagonal bisected

5. Non-congruent angles are bisected

- 1. 2 pairs of congruent base angles
- 2. Diagonals are congruent
- 3. One pair of parallel sides
- 4. Leg angles supplementary
- 5. Midsegment = ½ (b1 + b2)

Homework has ALL those characteristics!!!)

Quadrilateral Characteristics Summary

Convex Quadrilaterals

4 sided polygon

4 interior angles sum to 360

4 exterior angles sum to 360

Parallelograms

Trapezoids

Bases Parallel

Legs are not Parallel

Leg angles are supplementary

Median is parallel to basesMedian = ½ (base + base)

Opposite sides parallel and congruent

Opposite angles congruent

Consecutive angles supplementary

Diagonals bisect each other

Rectangles

Rhombi

IsoscelesTrapezoids

All sides congruent

Diagonals perpendicular

Diagonals bisect opposite angles

Angles all 90°

Diagonals congruent

Legs are congruent

Base angle pairs congruent

Diagonals are congruent

Squares

Diagonals divide into 4 congruent triangles

Homework has ALL those characteristics!!!)

In parallelogram PNWL, NW = 12, PM = 9, and mWLP = 144°. Find each measure.

1.PW2.mPNW

18

144°

Homework has ALL those characteristics!!!)

QRST is a parallelogram.

Find each measure.

a.TQb.mT

71°

28

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