Properties of Quadrilaterals 3.2

♥Any four sided polygon is a quadrilateral.

♥We’ll study special quadrilaterals in this section:

♥Trapezoid♥Parallelogram

♥Rhombus♥Rectangle

♥Square♥Kite

Properties of Parallelograms♥ Opposite sides of a

parallelogram are parallel♥ Opposite sides are congruent♥ Opposite angles of a

parallelograms are congruent.♥ Diagonals of a parallelogram

bisect each other♥ Consecutive angles of a

parallelogram are supplementary

♥ Alternate interior angles are congruent

supplementary

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

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

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

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

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

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

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

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

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°

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)

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:

3.2A and 3.2B

Section 9 - 41