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Lesson 6.1 Polygons. Today, we will learn to… > identify, name, and describe polygons > use the sum of the interior angles of a quadrilateral. triangle. quadrilateral. pentagon. hexagon. heptagon. octagon. nonagon. decagon. dodecagon. Theorem 6.1 Interior Angles of a Quadrilateral. - PowerPoint PPT Presentation
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Lesson 6.1Polygons
Today, we will learn to…> identify, name, and describe polygons > use the sum of the interior angles of a quadrilateral
# of Sides Name3
4
5
6
7
8
9
10
12
trianglequadrilateral
pentagonhexagonheptagon
octagonnonagon
decagon
dodecagon
Theorem 6.1Interior Angles of a
Quadrilateral
The sum of the measures of the interior angles of a quadrilateral is ______360°
Section 6.1 Vocabulary
ConvexConcave
EquilateralEquiangular
RegularDiagonal
Sides:
Vertices:
Diagonals:
S
T
U
DY
ST TU UD DY YS
S, T, U, D, Y
SU SD TD TY UY
S
T
U
DY
There are 10 possible names of this pentagon.
STUDYSYDUTTUDYSTSYDUUDYSTUTSYD
DYSTUDUTSYYSTUDYDUTS
How many diagonals can be drawn from N?
N M
O
PQ
R
Starting with N, give 2 names for the hexagon.
N M
O
PQ
R
NMOPQR NRQPOM
Is this a polygon? If not, explain. If so, is it
convex or concave?
Yes, it’s a convex
pentagon
Is this a polygon? If not, explain. If so, is it
convex or concave?
No, polygons must be made of
segments
Is this a polygon? If not, explain. If so, is it
convex or concave?
Yes, it’s a concave
dodecagon
Is this a polygon? If not, explain. If so, is it
convex or concave?
No, polygons must be closed
figures
Find x.
90 + 87 + 93 + x = 360x = 90
Find x.
3x + 3x + 2x + 2x = 360x = 36
Lesson 6.2Properties of Parallelograms
RULERS AND PROTRACTORS
Today, we will learn to…> use properties of parallelograms
A quad is a parallelogram if and only if two pairs of opposite sides are parallel
parallelogram
Draw a Parallelogram.
Measure each angle.Measure each side in centimeters.
Theorems 6.2-6.5If a quadrilateral is a parallelogram, then…
1) 6.22) 6.33) 6.44) 6.5
… opposite sides are __________congruent
… opposite angles are__________.congruent
… consecutive angles are__________.supplementary
1 2
34
m m m m
m m m m
1 2 180 1 4 180
3 2 180 3 4 180
… diagonals __________each other.
bisect
ABCD is a parallelogram. Find the missing angle and side measures.
1.A B
CD
105˚10
66
10
75˚
75˚
105˚
ABCD is a parallelogram. Find AC and DB.
2. A
CD
8
85
B
5
AC = 10 DB = 16
3. In ABCD, m C = 115˚. Find mA and mD.
4. Find x in JKLM.J K
LM(4x-9)˚
(3x+18)˚
mA = 115˚ mD = 65˚
x = 27
ABCD is a parallelogram.
EC =
m BCD =
m ADC =
AD =
5
8
70° 110°
The figure is a parallelogram.
x = y = 5 4
2x – 6 = 4 2y = 8
The figure is a parallelogram.
x = y = 30 6 4x + 2x = 180 2y + 3 = y + 9
The figure is a parallelogram.
x = y = 3 6
y
y
3x + 1 = 10 2y – 1 = y + 5
The figure is a parallelogram.
x = y = 40 8 3x – 9 = 2x + 31 4y + 5 = 2y + 21
Lesson 6.3Proving that Quadrilaterals
are Parallelograms
What is a converse?
Today, we will learn to…> prove that a quadrilateral is a
parallelogram
Theorem 6.6
If both pairs of opposite sides are __________,
then it is a parallelogram.congruent
Theorem 6.7If both pairs of opposite angles are __________,
then it is a parallelogram.congruent
Is ABCD a parallelogram? Explain.
1. 2.A B
CD
10
6
10
6
A B
CDyes
no
Theorem 6.8If an angle is
_______________ to both of its consecutive angles, then it is a parallelogram.
supplementary
1
2
3 m1 + m3 = 180˚m1 + m2 = 180˚
Theorem 6.9If the diagonals
__________________, then it is a parallelogram.
bisect each other
AE = ECand
DE = EB
A
D
B
C
E
Is ABCD a parallelogram? Explain.
3. 4. A B
CD
A B
CD
104˚
86˚ 104˚
no yes
Theorem 6.10If one pair of opposite sides are ___________
and __________, then it is a parallelogram.
congruentparallel
5.
8.
7.
6.
No Yes
Yes No
9. List 3 ways to prove that a quadrilateral is a parallelogram
1) prove that both pairs of opposite sides are __________
2) prove that both pairs of opposite sides are __________3) prove that one pair of opposite sides are both ________ and ________
parallel
congruent
parallel congruent
A ( , ) B ( , ) C ( , ) D ( , )
Prove that this is a parallelogram…
slope of AB isslope of BC isslope of CD isslope of AD is
0
4-2/5
-2/5
AB =BC =CD = AD =
4.15.44.15.4
2 3 4 -2 6 -3 2
4
Lesson 6.4Special
Parallelograms
Today, we will learn to…> use properties of a rectangle,
a rhombus, and a square
A square is a parallelogram with four congruent sides and four right angles.
A rhombus is a parallelogram with
four congruent sides.
A rectangle is a parallelogram with four right angles.four congruent sides. four right angles.
four congruent sides four right angles
parallelograms
rhombuses rectangles
squares
Sometimes, always, or never true?
1. A rectangle is a parallelogram.
2. A parallelogram is a rhombus.
3. A square is a rectangle.
4. A rectangle is a rhombus.
5. A rhombus is a square.
always true
sometimes true
always true
sometimes true
sometimes true
Geometer’s Sketchpad
mAEB = 90CD = 4.48 cmBC = 4.48 cmAD = 4.48 cmAB = 4.48 cm
E
C
A B
DWhat do we know about the diagonals in a
rhombus?
The diagonals of a rhombus are _____________.perpendicular
Theorem 6.11
What do we know about the diagonals in a rhombus?
mECD = 40
mEDA = 50 mEDC = 50
mEAD = 40 mEAB = 40
mECB = 40mEBC = 50 mEBA = 50
E
C
A B
D
The diagonals of a rhombus _____________________.bisect opposite angles
Theorem 6.12
What do we know about the diagonals in a rectangle?
ED = 4.51 cmEB = 4.51 cm
EC = 4.51 cmEA = 4.51 cm
E
C
A B
D
The diagonals of a rectangle are _____________. congruent
Theorem 6.13
6. In the diagram, PQRS is a rhombus. What is the value of y?
2y + 3
5y – 6
P Q
RS
y = 3
Find x. 7. rhombus
A
B
C
Dxº
52º
x = 38º
Find m CDB. 8. rhombus
A
B
C
D32º
mCDB =32º
Find AB.9. rectangle
A B
CD
10 12
AB = 16
?
202 = x2 + 122
10
Find x.10. square
A B
CD
xº xº
x = 45˚
Find EA & AB.11. square
EA =
A B
CD
4
EAB = 5.7
x2 = 42 + 42
x2 = 16 + 16x2 = 32x = 5.7
4
4
Lesson 6.5Trapezoids
& Kites
Today, we will learn to…> use properties of trapezoids
and kites
A trapezoid is a quadrilateral with only
one pair of parallel sides.
A B
D C
base
base
leg leg
B A
D
C
Compare leg angles.
Geometer’s Sketchpad
mC = 65mD = 115mA = 90mB = 90
In ALL trapezoids, leg angles are
_______________supplementary
A trapezoid is an
isosceles trapezoid
if its legs are congruent.
Geometer’s Sketchpad
Compare base angles.Compare leg angles.How do you know it is isosceles?
mA = 67 mD = 67 mC = 113 mB = 113 CD = 3.7 cmAB = 3.7 cm
A D
B C
Theorem 6.14 & 6.15A trapezoid is isosceles if and
only if base angles are ___________.congruent
Base angles are congruent.
A B
CDAC BD
The trapezoid is isosceles.
The triangles share CD.ADC BCD by SAS
CPCTC
Theorem 6.16A trapezoid is isosceles if
and only if its diagonals are __________.congruent
AC BD
A B
CD
ABCD is an isosceles trapezoid. Find the missing angle measures.
1. A B
CD100°
80° 80°
100°
2. The vertices of ABCD are A(-1,2), B(-4,1), C(4,-3), and D(3,0). Show that ABCD is an isosceles trapezoid.
Figure is graphed on next slide.
3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
D(3, 0)
C(4, -3)
B(-4, 1)
A(-1, 2)
AD || BC ?
AB =CD =
- ½ - ½
Legs are ? Diagonals are ? AC=BD =
50 10 10 50
OR?
Slope of AD isSlope of BC is
x = 118 Find x.
The midsegment is a segment that connects the midpoints of
the 2 legs of a trapezoid.
Geometer’s SketchPad
EF = 8 cmCD = 12 cm
AB = 4 cm
EF = 7 cmCD = 11 cm
AB = 3 cm
A
EF = 5 cmCD = 6 cm
AB = 4 cm
EF = 7 cmCD = 9 cm
AB = 5 cm
FE
A B
D C
Theorem 6.17Midsegment Theorem for
TrapezoidsThe midsegment of a
trapezoid is _________ to each base and its length is ______________ of the
bases.
parallel
the average
Find x.
3. 4.
7
11
x
x
17
20
x = 9 x = 23
KITE
A kite has two pairs of consecutive congruent
sides but opposite sides are not congruent and no sides
are parallel.
Kite
What do we know if these points are equidistant from the endpoint of the segment?
Theorem 6.18
In a kite, the longer
diagonal is the _________________
of the shorter diagonal.perpendicular bisector
Kite
What do we know about congruent triangles?
How do we know the triangles are congruent?
Kite
Theorem 6.19In a kite, exactly one pair of opposite angles
are ________.congruent
The congruent angles are formed by the noncongruent sides.
Find x and y.
5. 6.
5
x yx˚ 125˚
y˚
(y+30)˚29
x = 2 y = 2
x = 125
y = 40
Theorem 6.19*
In a kite, the longer diagonal
________________.bisects opposite angles
mJ =70°
mL = 70°
Find the missing angles.
x =35
Find x.
Find x.
x = 110
Find x.
x = 5
Based on our theorems, list all of the properties that must be true for the quadrilateral.
1. Parallelogram (definition plus 4 facts)
2. Rhombus (plus 3 facts)
3. Rectangle (plus 2 facts)
4. Square (plus 5 facts)
Parallelogram
1) opposite sides are parallel
2) opposite sides are congruent
3) opposite angles are congruent
4) consecutive angles are supplementary
5) diagonals bisect each other
Rhombus1) equilateral2) diagonals are perpendicular3) diagonals bisect opposite angles
Rectangle1) equiangular2) diagonals are congruent
Square1) equilateral2) equiangular3) diagonals are perpendicular4) diagonals bisect opposite angles5) diagonals are congruent
Lesson 6.6Identifying Special
Quadrilaterals
Complete the chart of characteristics of special quadrilaterals.
Today, we will learn to…> identify special quadrilaterals
with limited information
Given the following coordinates, identify the quadrilateral.
(-2, 1)(-2, 3)(3, 6) (0, 1)
kite
Given the following coordinates, identify the quadrilateral.
(0, 0)(4, 0)(3, 7) (1, 7)
trapezoid
Given the following coordinates, identify the quadrilateral.
rectangle
(-1, -3)(4, -3)(4, 3) (-1, 3)
Given the following coordinates, identify the quadrilateral.
rhombus
(-2, 0)(3, 0)(6, 4) (1, 4)
In quadrilateral WXYZ, WX = 15, YZ = 20, XY = 15,
ZW = 20. What is it?
It is a kite!
Lesson 6.7Areas of Triangles and
Quadrilaterals
Today, we will learn to…> find the area of triangles and
quadrilaterals
Postulate 22Area of a Square
Area = side2
A=s2
Postulate 23Area Congruence Postulate
If two polygons are congruent, then they have the same area.
Theorem 6.20Area of a Rectangle
Area = base ( height )
A = bh
1. Find the area of the polygon made up of rectangles.
4 m
10 m
2 m
9 m
11 m
7 m11(2) = 22 m2
8(4) =
32 m2
5(4)= 20 m2
74 m2
?
??
Postulate 24
Area Addition Postulate
The area of a region is the sum of
the areas of its nonoverlapping
parts.
Theorem 6.21Area of a Parallelogram
Area = base ( height)
A=bh
Do experiment.
Theorem 6.22Area of a Triangle
A=½ bh
Area of a Trapezoid
hh
b2
A = ½ h b1 + ½ h b2
b1
A = ½ h (b1 + b2)
A = ½ h b1 + ½ h b2
Theorem 6.23Area of a Trapezoid
A = ½ height (sum of bases)
A=½ h (b1+b2)
2. parallelogram 3. trapezoid
6
4 55 5
3
4
9
A = 6(4)
A = 24 units2A = ½ 4(9+3)
A = 24 units2
Area of a Kite
b
b
x
y
A = ½ bx + ½ by
A = ½ b (x + y)What is b? a diagonal
What is x + y? a diagonal
A = ½ d1 d2
Theorem 6.24Area of a Kite
Area = ½ (diag.)(diag.)
A=½ d1 d2
Area of a RhombusA = ½ bx + ½ by
A = ½ b(x + y)What is b? a diagonal
What is x + y? a diagonal
A = ½ d1 d2
b
b
x
y
Theorem 6.25Area of a Rhombus
Area = ½ (diag.)(diag.)
A=½ d1 d2
4. Rhombus 5. Kite
4
35
34
A = ½ 6(8)
A = 24 units2
A = ½ 6(9)
A = 27 units2
6. Rhombus 7. Trapezoid
8
x
A = 80 units2
x = 5
A = 55 units2
h = 5
h
13
9
8. Find the total area.
15
8 A = ½(10)(8+20)
A = 440 units2
20
25A = 140
A = 20(15)
A = 300
?10
A = 12(11)
blue A = ½ (12)(5)
11
12
A = 132
132 = 122 + x2
x = 513
just blue?
blue A = 30
pink A = 132 – 60
pink A = 72
2 blue regions A = 60
?5
9. Find the areas of the blue and pink regions.