Math 8: Geometry

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Math 8 Geometry

• ORDER INTELLIGENCE DESIGN

Pentagon Building
Nautilus Shell
Solar System
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Identifying Shapes recall the name click
Number of Sides 3
4 5 6
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Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
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Angle Types

• There are four angle types you need to know.
• Name them and their definitions.

Acute
Between 0 and 90
Obtuse
Between 90 and 180
Right
90 degrees
Straight
180 degrees
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Label Each Angle Type
Acute
Right
Obtuse
Straight
What's the other name for a "right" angle?
Lindquist Angle!
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A Few More Questions
What is the largest acute angle that you could
double and still get an acute angle?
44. If you tried 45 and doubled it you get 90
a rt lt.
What is the smallest acute angle you could triple
and get an obtuse angle?
Well, we need to find the smallest number that
tripled is bigger than 90. Since 30 x 3 90,
the angle we are looking for is 31. 31 x 3
93.
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A Few Questions .

• 1) An angle that measures 54 degrees is
  _________.
• a) Acute b) Obtuse c) Straight d) Right

• A 180 degree angle is a __________ angle.
• a) Acute b) Obtuse c) Straight d) Right

3) Perpendicular lines make what kind of angles?
Right Angles!!!!!
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Problems to Ponder
Two congruent acute angles add up to a right
angle. How many degrees in each angle?
45 degrees. 45 45 90
Is it possible for a quadrilateral to have no
acute angles?
Sure ? ! A rectangle has 4 right angles.
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Triangle Types Choose ALL possible choices from
the words belowScalene Equilateral Acute
Right Isosceles Obtuse

• This triangle has all sides equal.
• 2) This triangle has no sides equal and all
  angles less than 90 degrees.
• 3) This triangle has two equal sides.
• 4) This triangle has an angle greater than 90 and
  less than 180 degrees.
• 5) The angles of this triangle are 90, 45 and 45.

Equilateral
Scalene. Acute.
Isosceles
Obtuse
Right. Isosceles.
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How many degrees in the 3 lts of a
triangle?_______
180
How many degrees in the missing angle below?
58 52 110 180 110 70
58
70
52
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Find the value of n
This is isosceles, so it has two congruent angles
as well as two congruent sides. What does the
base angle on the left equal?
48
n like the one on the right ?
Therefore .. 2n 48 180. Solve this!
n
2n 48 180 - 48 -48 2n
132 n 66 ? !
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• The angles of a triangle measure n, 2n and 3n.
• Find n.
• Is the triangle best classified as acute, right
  or obtuse?

3 angles add up to 180 degrees!
n 2n 3n 180 6n 180 n 30
30
n 30 2n 60 3n 90
60
90
It's a Right Triangle!
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Find the value of n
The vertex angle of an isosceles triangle
measures 80. Find the measure of each of the
base angles.
80
The three angles sum to 180, so N N 80 180
n
n
2N 80 180 - 80 -80 2N
100 N 50
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Tricky Triangle Questions
Johnny says an equilateral triangle can not be
isosceles. Is Johnny right or wrong?
Johnny is wrong. An isosceles triangle has two
sides equal. Two of the sides of the equilateral
triangle ARE equal. It does not matter that the
third sides is, too.
Is it possible for a triangle to have two right
angles?
No. Numerical explanation two of the angles
would already sum to 180 (90 90). There would
be no room for a third angle. Geometric
explanation Two sides of the triangle would
never meet.
90
90
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Angle Triangle Questions
1) An acute triangle has three acute angles.
Does an obtuse triangle have 3 obtuse angles?
Why or why not?
No! The three angles of a triangle sum to 180.
All obtuse angles are greater than 90, so three
of them would sum to 270 or more.
2) True or false and why An obtuse triangle
can never be isosceles?
False! One example is a triangle with angles of
100, 40 and 40. Also, see the diagram below.
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The Quadrilateral Family

• Papa Parallelogram

Opposite Sides Parallel Equal Opposite Angles
Equal Angles Sum to 360
Randy Rectangle
Rebecca Rhombus
Parallelogram 4 Equal Sides
Parallelogram 4 Rt lts Congruent Diagonals
Squiggy Square
Everything !!!!!
What is important about each?
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The Tricky Trapezoid
What makes a trapezoid tricky?
110
110
It is kind of like a parallelogram but it only
has one pair of parallel sides! A parallelogram
has two pairs.
70
70
How is a trapezoid like a parallelogram?
The interior angles add up to 360. Check out the
angles..
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Quadrilateral QuestionsAlways. Sometimes.
Never.
A rectangle is a parallelogram.
Always the opposite sides of a rectangle are
congruent.
A rhombus is a square.
Sometimes. But a rhombus does not have to have a
right angle and a square does.
A trapezoid is a parallelogram.
Never! A trapezoid only has one pair of parallel
sides.
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Quadrilateral QuestionsAlways. Sometimes.
Never.
A parallelogram has 4 congruent sides. The
opposite angles of a square are each 100
degrees. The sum of the angles of a
parallelogram is 360.
Sometimes. When it does, it is a special type of
parallelogram called a rhombus.
rhombus
Never. They are each 90 degrees.
Absolutely! Think of the 4 rt angles of a
square. 90 x 4 360
90
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Quadrilateral Question
Two consecutive sides of a rhombus are 3b 5 and
b 11. Find b. Find the length of a side.
All Sides of a rhombus are equal!

• Strategies
• Diagram
• Equation
• Solve

3b 5 b 11
3b - 5
-b -b 2b 5 11 5 5
2b 16 b 8
b 11
Side 3x8 5 19
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Quadrilateral Question
Two opposite sides of a parallelogram measure 4(x
5) and 28. Find the value of x.
Opposite sides of a parallelogram are equal!
4(x 5)
28
4(x 5) 28 DISTRIBUTE! 4x 20 28 20
20 4x 48 x 12
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Quadrilateral Word Problems
The measure of an angle of a rectangle can be
expressed as 2x 10. Find x.

• Strategies
• Diagram
• Equation
• Solve

2x 10
The angles of a rectangle are right angles 90
2x 10 90 - 10 -10 2x 80
x 40
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Match the Name to the Pic

• 1) Rectangle only
• 2) Parallelogram
• 3) Square
• 4) Rhombus

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Key Terms Key Pictures
1) Define Complementary.
Two lts add up to 90
2) Define Supplementary
Two lts add up to 180
What is the trick for remembering that
complementary means 90 and supplementary means
180?
C comes before S, 90 before 180
3) What is the complement of 40?
50
4) What is the supplement of 100?
80
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Find the complement and supplement of an angle of
40 degrees.
Complement 50 Supplement 140.
What is the difference between the supplement and
complement of 40?
90
Will the difference between the complement and
supplement of any given angle always be 90? Why
or why not?
Yes! Think about it Complementary adds up to
90 and supplementary to 180. The difference
between those two numbers is 90 ? .
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Complementary or Supplementary?
Which picture is which and find the value of n.
Supplementary 2n 68 180 n 56
2n
68
Complementary 8 3n 66 90
63
3n - 3
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Pairs of Angles
Find the value of n in each diagram.
2n 43
7n 8
3n - 17
2n 12
These angles are called ________ and they are
_______.
These angles are called ___________ and they sum
to ______.
supplementary
vertical
equal
180

• 7n 8 2n 43
• 2n -2n
• 5n 8 43
• - 8 - 8
• 5n 35 n 7

3n 17 2n 12 180 5n 5 180 5
5 5n 185 n 37
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Perimeter and Area
Perimeter measures the _____________ of a
figure. Area measures the _____________ of a
figure.
outside
10 in
32 in
inside
adds
multiplies
Which operation?
Perimeter ______. Area __________.
Find the perimeter and area of the rectangle
above.
Perimeter Area
Add all sides. 84 inches 10 in32 in10 in 32in
Area length x width 10 in x 32 in 320 sq. in.
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Circles Circumference Area
Circumference is like perimeter but only
applies to a circle. Area doesnt change it is
all about how much is inside.
What is the trick to remember the formulas?
Cherry Pie is delicious. Apple pies are,
too. C pd A pr2
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Parts of a Circle
For our purposes, there are two main parts of a
circle we need to know .. diameter and
radius. What is the difference?
The diameter goes all the way across. The radius
only goes halfway.
diameter
radius
If the diameter if a circle is 18 inches, what is
the radius?
The radius only goes halfway across so it must be
half of 18 inches 9 inches.
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Find the Circumference and Area of the circle.
C p d (d is diameter)
10 m
C 3.1416 x 20 m C 62.832 m
A p r2 (r is radius)
A 3.1416 x 10 m x 10 m A 314.16 m2
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Two Parallel LinesCut by a Transversal
When two parallel lines are cut by a transversal,
many pairs of congruent and supplementary angles
result.
Notice that there are two different types of
angles in the diagram acute and obtuse.
For all practical purposes, all acute angles are
congruent to each other as are all obtuse
angles. Any time an acute and an obtuse are
paired, they are supplementary!
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Parallel Lines Angles
Name all of the acute angles in the diagram.
lt1, lt4, lt5, lt8
1
2
3
4
What do you know about these angles?
They are all equal.
5
6
Name all of the obtuse angles in the diagram.
8
7
lt2, lt3, lt6, lt7
What do you know about these angles?
They are all equal.
If a question pairs an acute angle with an obtuse
angle, they are _________.
supplementary
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Parallel Lines Angles
Parallel Lines Angles
Name all of the angles in the diagram that are
congruent to lt1. Once you select them, click and
they will appear light blue.
1
2
3
4
5
6
8
7
Name all of the angles in the diagram that are
congruent to lt7. Once you select, click and they
will appear red.
lt2 and lt8 are ______________.
supplementary
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Parallel Lines Angles
If mlt5 70, find the measures of all the other
angles mlt1 mlt2 mlt3 mlt4 mlt6 mlt7 mlt8
1
2
3
4
70
5
6
110
8
7
110
70
110
110
70
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Parallel Lines Angles
If mlt5 75, find the measures of all the other
angles mlt1 mlt2 mlt3 mlt4 mlt6 mlt7 mlt8
1
2
3
4
75
5
6
105
8
7
105
75
105
105
75
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Parallel Lines Angles
If mlt4 3x 18 and mlt5 x 24, find the mlt4.
1
2
3
4
Equal or Supplementary?
Find x.
Substitute
5
6
lt4 lt5 are both acute, so they are equal. 3x
18 x 24
8
7
-x -x 2x 18 24 18
18 2x 42 x 21
mlt4 3(21) 18 63 18 45
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Parallel Lines Angles
If mlt4 2x 20 and mlt6 3x 10, find x.
1
2
3
4
lt4 is acute and lt6 is obtuse, so the two are
supplementary!
5
6
8
7
2x 20 3x 10 180 (since they are on the
same side, like terms go together) 5x 30 180
- 30 -30 5x 150 x
30
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Surface Area of a Rectangular Prism
How many faces does this prism have?
6
Ill name one side. You name its opposite partner.
FRONT
LEFT
TOP
BACK
RIGHT
BOTTOM
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Finding The Surface Area
Step 1 Find the fronts area.
TOP
5 in x 8 in 40 in2 x 2 80 in2
FRONT
5 in
Why double the 40?
The Back!
2) Find the top area and double it.
8 inches
4 in
8 in x 4 in 32 in2 x 2 64 in2
RIGHT
3) Find the area of the right and double it
because of the left partner or equal face.
4 in x 5 in 20 in2 x 2 40 in2
Add the areas of the faces to get the surface
area)
80 in2 64 in2 40 in2 184 in2
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Surface Area of a Cylinder
What shapes do you get when you unfold a cylinder?
Two congruent circles on the top and bottom and a
rectangle.
If the width of the rectangle on the side is the
height, how do you find the length?
Radius 10 Height 25
It is the circumference of the circles.
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Computing the Surface Area
The reference sheet gives the following formula
for the surface area of a cylinder SA 2prh
2pr2
Substitute and crank out the antza ? . SA
2(3.1416)(10)(25) 2(3.1416)(102)
1570.8 628.32 2199.12
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Shaded Areas
Find the area of the shaded region.
6 in
5 in
8 in
Area of green rectangle A l x w 8in x
25in200in2
8 in
8 in
25 in
Area of the triangle A ½ b h ½ (6 in) (8
in) 24 in2
Area of the parallelogram b x h (5 in) (8 in)
40 in2
Area of the shaded region is 200 in2 (24 in2
40 in2) 136 in2
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WINK What I Need To Know
Polygon Types
Complemetary Supplementary Vertical Angles
Parallel Lines
Geometry
Perimeter Area Circumference
Angle Types
Surface Area