ch 6 quadrilaterals
TRANSCRIPT
POLYGONS
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°
S
T
U
DY
There are 10 possible names of this pentagon.
STUDYSYDUTTUDYSTSYDUUDYSTUTSYD
DYSTUDUTSYYSTUDYDUTS
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?
No, polygons must be closed
figures
Lesson 6.2Properties of Parallelograms
RULERS AND PROTRACTORS
Today, we will learn to…> use properties of parallelograms
… 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
ABCD is a parallelogram. Find the missing angle and side measures.
1.A B
CD
105˚10
66
10
75˚
75˚
105˚
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
Lesson 6.3Proving that Quadrilaterals
are Parallelograms
What is a converse?
Today, we will learn to…> prove that a quadrilateral is a
parallelogram
Theorem 6.8If 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
Theorem 6.10If one pair of opposite sides are ___________
and __________, then it is a parallelogram.
congruentparallel
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
ParallelogramsToday, 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
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
mÐAEB = 90°CD = 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?
What do we know about the diagonals in a rhombus?
mÐECD = 40°
mÐEDA = 50°mÐEDC = 50°
mÐEAD = 40°mÐEAB = 40°
mÐECB = 40°mÐEBC = 50°mÐEBA = 50°
E
C
A B
D
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
Geometer’s Sketchpad
Compare base angles.Compare leg angles.How do you know it is isosceles?
mÐA = 67°mÐD = 67°mÐC = 113°mÐB = 113°CD = 3.7 cmAB = 3.7 cm
A D
B C
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
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
Geometer’s SketchPad
EF = 8 cmCD = 12 cm
AB = 4 cm
EF = 7 cmCD = 11 cm
AB = 3 cm
AEF = 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
KITE
A kite has two pairs of consecutive congruent
sides but opposite sides are not congruent and no sides
are parallel.
Theorem 6.18
In a kite, the longer
diagonal is the _________________
of the shorter diagonal.perpendicular bisector
Theorem 6.19In a kite, exactly one pair of opposite angles
are ________.congruent
The congruent angles are formed by the noncongruent sides.
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
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.
rectangle
(-1, -3)(4, -3)(4, 3) (-1, 3)
Lesson 6.7Areas of Triangles and Quadrilaterals
Today, we will learn to…> find the area of triangles and
quadrilaterals
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.
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
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
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.