Properties of Quadrilaterals 3.2

♥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

Properties of Parallelograms♥ 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

|| ||

Alternate interior

a. mMJK b. mJML

c. mJKL d. mKJL

e. a f. b

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

7

80

100

80

30

21

a. mPLM

b. mLMN

c. d =

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

108

72

11

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

x = 12 y = 21a. x b. y

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

x = 7 y = 4a. x b. y

Find the value of x that makes the figure a parallelogram. State the property.

x = 46a. x

Find the values so that the figure is a parallelogram State the property.

x = 25 y = 15 a = 7 b = 7

x = 8 y = 65 w = 4 z = 4½

a. x b. y c. a d. b

e. x f. y g. w h. z

a. mMNP

b. mNRP

c. mRNP

d. mRMN

e. mMQN

f. mMQR

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

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

109

83

8 6.45 3.525 6.13

97

38

71

33

AssignmentGeometry:

Properties of a Parallelogram

Properties of a Rhombus (Rhombi)♥ 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

Rhombus

NMa.

Find the indicated measure in rhombus JKLMKM = 8 and JL = 6. State the property.State the property.

4

JM d. 5

m KNL b. 90°

m KJL e. 53°

JNc. 3

m KJM f. 106°

37

Properties of Rectangles♥ 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 congruent and they bisect each other

Rectangles

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

If mMNL = 140 answer the following?

x = 8

a.a. mmJNK JNK

b. mb. mMNJ MNJ

c. mc. mLNKLNK

d. md. mMJKMJK

e. me. mNLK NLK

f. mf. mNLM NLM

140°

40°

90°

40° 70°

20°

70°

20°

g. mg. mLJK LJK

h. mh. mLJMLJM

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

y = 9a. x b. y c. za. x b. y c. zx = 5 z = 12.5

(2z)

+ 11)

WXYZ is a rectangle. WXYZ is a rectangle.

Find each measure Find each measure

if if mm1 = 351 = 35. .

State the property.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

35° 55°

55°

35°

35° 55°

55°

35°

70° 70° 110° 110°

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.

Quadrilateral Quadrilateral JKMN JKMN is a is a rectangle. rectangle. Find each Find each measure. measure. State the property.State the property.

36

11

8 or 24

27

37

3

Television screens are rectangles and are measured by their diagonals.

Find the length of the diagonal.

21² + 36² = c²1737 = c²

c = 1737

c 41.6773

a² + b² = c²

in.

Properties of Squares♥ 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

Parallelogram ABCD is a square.Find x and y.

10 in.

A B

C Dy 14.14

a.a. x x

b. yb. y

10² + 10² = c²200 = c²c = 200c 14.14

x = 45

a² + b² = c²

AssignmentGeometry:

Rectangles, Rhombus & Squares

Inheritance of Properties

Kites TrapezoidsIsoscelesTrapezoid

Properties of a Kite:A quadrilateral with NO parallel sides.

♥ 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

Kites

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

x + 4

14

y + 16

2x + 12

16

a.a. x x

b. yb. y

10

c. IT c. IT

d. KEd. KE 32

14

Find the value of x and y in the kite below.

a.a. x b. yx b. y

24² + (SO)² = 27²

(SO)² = 153 SO = 153SO 12.4

a² + b² = c²

576 + (SO)² = 729

2x + 5y = 12.46 + 5y = 12.45y = 6.4y = 1.28

12.4

4x + 3 = 154x = 12x = 3

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

a.m1 b. m2 c. m3 d. m4 e. m5 f. m6 g. m7

27° 52°

63°

38°

38° 63°

90°

27521

2

34

56

7

Trapezoid

Isosceles Trapezoid

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.

Median

Base

Base

For isosceles trapezoid XYZW, Find the length of the median, mX and mZ.

6

1865

a.Median

b. mZ

c. mX

12

115°

65°

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

a. AB b. mQ c. mS16 60° 135°

PQRS is an isosceles trapezoid; find x.

a = 9 or a = –9

2a2 – 54 = a2 + 27

a2 = 81

XY is the midsegment of trapezoid ABCD; find x. 17x

22.5x + 9

30x + 12

9x5.222

12x47

9x5.226x5.23

x = 3

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

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)

Quadrilateral Characteristics SummaryConvex Quadrilaterals

Squares

RhombiRectangles

Parallelograms Trapezoids

IsoscelesTrapezoids

Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other

Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)

Angles all 90°Diagonals congruent

Diagonals divide into 4 congruent triangles

All sides congruentDiagonals perpendicularDiagonals bisect opposite angles

Legs are congruent Base angle pairs congruent Diagonals are congruent

4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360

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

1. PW 2. mPNW

18 144°

QRST QRST is a parallelogram. is a parallelogram. Find each measure.Find each measure.

a. TQ b. mT

28 71°

AssignmentGeometry:

Trapezoids & Kites

AssignmentGeometry:

3.2A and 3.2B

Section 9 - 41