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