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
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Alternate interior
Find a and b so that the quadrilateral
is a parallelogram State the property.
a. mMJK 100
b. mJML
80
c. mJKL
80
d. mKJL
30
e. a
7
f. b
21
Find d so that the quadrilateral is a
parallelogram. State the property.
a. mPLM
108
b. mLMN
72
c. d = 11
Find x and y so that the quadrilateral is a
parallelogram State the property.
a. x
x = 12
b. y
y = 21
Find x and y so that the quadrilateral is a
parallelogram. State the property.
a. x
x=7
b. y
y=4
Find the value of x that makes the figure a
parallelogram. State the property.
a. x
x = 46
Find the values so that the figure is
a parallelogram State the property.
a. x
b. y
c. a
d. b
x = 25
y = 15
a=7
b=7
e. x
f. y
g. w
h. z
x=8
y = 65
w=4
z = 4½
Find x, y, w, and z so that the quadrilateral
is a parallelogram. State the property .
a. mMNP 71
b. mNRP 33
c. mRNP 38
d. mRMN 109
e. mMQN 97
f. mMQR 83
g. x 8
h. y 6.45
i. w 3.525
j. z 6.13
Assignment
Geometry:
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
Find the indicated measure in rhombus JKLM
KM = 8 and JL = 6. State the property.
a. NM 4
b. m
c.
KNL 90°
JN 3
d. JM 5
e. m
KJL 53°
f.
KJM 106°
m
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?
x=8
If mMNL = 140
answer the following?
a. mJNK 140°
d. mMJK 90°
g. mLJK 20°
b. mMNJ 40°
e. mNLK 70°
h. mLJM 70°
c. mLNK
f. mNLM 20°
40°
In rectangle ABCD shown below, find the
value of x, y, and z. State the property.
(2z)
+ 11)
a. x
b. y
c. z
x=5
y=9
z = 12.5
WXYZ is a rectangle.
Find each measure
if m1 = 35.
State the property.
a.m1 35° b. m2
55°
c. m3
e. m5 35° f. m6
55°
g. m7 55° h. m8
i. m9
70°
55°
d. m4 35°
35°
j. m10 70° k. m11 110° l. m12 110°
Quadrilateral JKMN is a
rectangle. Find each
measure.
State the property.
a. If NQ = 5x + 3 & QM = 4x + 6, find NK. 36
b. If NQ = 2x + 3 & QK 5x - 9, find JQ. 11
c. If NM = 2x + 14 & JK = x2 - 1, find JK. 8 or 24
d. If mNJM = 2x + 3 & mKJM = x + 6, find x. 27
e. If mNKM = x2 + 4 & mKNM = x + 30, find mJKN. 37
f. If mJKN = 16x & mNKM = 14x, find x. 3
Television screens are rectangles and
are measured by their diagonals.
Find the length of the diagonal.
a² + b² = c²
21² + 36² = c²
in.
1737 = c²
c =  1737
c  41.6773
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.
A
C
B
a² + b² = c²
10² + 10² = c²
200 = c²
10 in. c =  200
c  14.14
D
a. x x = 45
b. y y  14.14
Assignment
Geometry:
Rectangles, Rhombus & Squares
Inheritance of Properties
Kites
Trapezoids
Isosceles
Trapezoid
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 mK  mT)
♥ 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
a. x 10
14
b. y 16
c. IT 14
y + 16
d. KE 32
2x + 12
Find the value of x and y in the kite below.
12.4
a² + b² = c²
24² + (SO)² = 27²
576 + (SO)² = 729
(SO)² = 153
SO =  153
SO  12.4
a. x
4x + 3 = 15
4x = 12
x=3
b. y
2x + 5y = 12.4
6 + 5y = 12.4
5y = 6.4
y = 1.28
In kite ABCD, find the measures of the
numbered angles.
6 2
5
4
3
1
27
52
7
a.m1 27° b. m2
52°
c. m3
e. m5 38° f. m6
63°
g. m7 63°
90°
d. m4 38°
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.
Base
Median
Base
For isosceles trapezoid XYZW, Find the
length of the median, mX and mZ.
6
a.Median
12
b. mZ 115°
65
18
c. mX
65°
In trapezoid QRST, A and B are midpoints
of the legs. Find AB, mQ, and mS.
a. AB
16
b. mQ
60°
c. mS
135°
PQRS is an isosceles trapezoid; find x.
2a2 – 54 = a2 + 27
a2 = 81
a = 9 or a = –9
XY is the midsegment of trapezoid ABCD; find x.
17x
22.5x + 9
30x + 12
47 x  12
 22 . 5 x  9
2
23 . 5 x  6  22 . 5 x  9
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 Summary
Convex Quadrilaterals
Parallelograms
4 sided polygon
4 interior angles sum to 360
4 exterior angles sum to 360
Opposite sides parallel and congruent
Opposite angles congruent
Consecutive angles supplementary
Diagonals bisect each other
Rectangles
Trapezoids
Bases Parallel
Legs are not Parallel
Leg angles are supplementary
Median is parallel to bases
Median = ½ (base + base)
Rhombi
Angles all 90°
Diagonals congruent
All sides congruent
Diagonals perpendicular
Diagonals bisect opposite angles
Squares
Diagonals divide into 4 congruent triangles
Isosceles
Trapezoids
Legs are congruent
Base angle pairs congruent
Diagonals are congruent
In parallelogram PNWL, NW = 12, PM = 9,
and mWLP = 144°. Find each measure.
1. PW
18
2. mPNW
144°
QRST is a parallelogram.
Find each measure.
a. TQ
28
b. mT
71°
Assignment
Geometry:
Trapezoids & Kites
Assignment
Geometry:
3.2A and 3.2B
Section 9 - 41