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Properties of Quadrilaterals 3.2

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Properties of Quadrilaterals 3.2

Homework

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

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Homework

Homework

Homework

Alternate interior

Homework

Homework

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

Homework

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

- mPLM
- b. mLMN
- c. d =

108

72

11

Homework

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

a. xb. y

y = 21

x = 12

Homework

a. xb. y

x = 7

y = 4

Homework

a. x

x = 46

Homework

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

6.13

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

Assignment

Geometry:

Properties of a Parallelogram

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

Homework

Rhombus

a.

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

Homework

Rectangles

Homework

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)

+ 11)

Homework

a. x b. y c. z

z = 12.5

y = 9

x = 5

Homework

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

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.

c = 1737

Homework

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

Homework

A

B

10 in.

c = 200

C

D

Homework

a² + b² = c²

10² + 10² = c²

200 = c²

c 14.14

- x
- b. y

x = 45

y 14.14

Assignment

Geometry:

Rectangles, Rhombus & Squares

Homework

Kites

Trapezoids

Homework

Isosceles

Trapezoid

Homework

2 pair of consecutive congruent sides

Opposite sides are NOT congruent

Angles are congruent as marked

(also mK mT)

Diagonals are perpendicular

Notice only ONE diagonal

is bisected

Homework

Homework

x + 4

14

y + 16

2x + 12

Homework

- x
- b. y

10

16

c. IT

d. KE

14

32

Homework

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

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

Trapezoid

Homework

Isosceles Trapezoid

Homework

- 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

65

18

Homework

- Median
- b. mZ
- c. mX

12

115°

65°

Homework

a. AB b. mQ c. mS

135°

60°

16

Homework

PQRS is an isosceles trapezoid; find x.

2a2 – 54 = a2 + 27

a2 = 81

a = 9 or a = –9

17x

22.5x + 9

30x + 12

Homework

XY is the midsegment of trapezoid ABCD; find x.

x = 3

Homework

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

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

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

1.PW2.mPNW

18

144°

Homework

QRST is a parallelogram.

Find each measure.

a.TQb.mT

71°

28

Assignment

Geometry:

Trapezoids & Kites

Assignment

Geometry:

3.2A and 3.2B

Section 9 - 41