Transcript Symmetry in Polygons - Hazlet Township Public Schools

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
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
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
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
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
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
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
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
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, 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°
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 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:
3.2A and 3.2B
Section 9 - 41