Welcome to MM150 Unit 6
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Transcript of Welcome to MM150 Unit 6
Welcome to MM150 Unit 6
Seminar
• Line AB AB– a set of points with arrows on both ends means that it
extends in both directions to infinity
• Ray AB AB– has an endpoint and one end goes to infinity
• Line segment AB AB– Part of a line between two points, including the endpoints
• Open Line Segment AB– set of points on a line, between two points, excluding the
end points
AngleTwo rays that come together at a vertex
A
D
F
Vertex
Side
Side
Angle Measures
Right Angle 90 degrees
Straight Angle 180 degrees
Acute Angle 0 degrees < acute < 90 degrees
Obtuse Angle 90 degrees < obtuse < 180 degrees
More Angle Definitions
B D H
L M
2 angles in the same plane are adjacent angles if they have a common vertex and a common side, but no common interior points. Example: [ang]BDL and [ang]LDM Non-Example: [ang]LDH and [ang]LDM
2 angles are complementary angles if the sum of their measures is 90 degrees.Example: [ang]BDL and [ang]LDM
2 angles are supplementary angles if the sum of their measures is 180 degrees.Example: [ang]BDL and [ang]LDH
If the measure of [ang]LDM is 33 degrees, find the measures of
the other 2 angles.
B D H
L M
Given information:[ang]BDH is a straight angle[ang]BDM is a right angle
If [ang]ABC and [ang]CBD are complementary and [ang]ABC is 10 degrees less than [ang]CBD, find the measure of both angles.
B A
CD
[ang]ABC + [ang]CBD = 90Let x = [ang]CBDThen x – 10 = [ang]ABC
X + (x – 10) = 902x – 10 = 902x = 100X = 50 [ang]CBD = 50 degreesX – 10 = 40 [ang]ABC = 40 degrees
Vertical Angles
• When two straight lines intersect, the nonadjacent angles formed are called vertical angles. Vertical angles have the same measure.
2
1 3
4
< 1 = < 3
< 2 = < 4
8
Parallel Lines Cut by a Transversal 1 2
3 4
5 6
7 8
When two lines are cut by a transversal,
1.) alternate interior angles have the same measure (<3,<6; <4,<5)
2.) corresponding angles have the same measure (<1,<5; <2,<6; <3,<7; <4,<8)
3.) alternate exterior angles have the same measure (<1, <8; <2,<7)
* Vertical angles
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Example
1 2
3 4
5 6
7 8
If the measure of <1 is 45 degrees, find the remaining measures.
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Triangles• Isosceles Triangle – 2 equal sides and 2 equal
angles
• Equilateral Triangle – three sides equal and three angles equal
• Scalene Triangle – No two sides are equal in length
* All three angles of a triangle add up to 180 degrees. 11
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Similar Figures
A
B
C X
Y
Z
80[deg]
80[deg]
50[deg] 50[deg]50[deg]50[deg]
[ang]A has the same measure as [ang]X[ang]B has the same measure as [ang]Y[ang]C has the same measure as [ang]Z
XY = 4 = 2AB 2
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1 2
4 4
YZ = 4 = 2BC 2
XZ = 2 = 2AC 1
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Page 238 # 73
• Steve is buying a farm and needs to determine the height of a silo. Steve, who is 6 feet tall, notices that when his shadow is 9 feet long, the shadow of the silo is 105 feet long. How tall is the silo?
6 ft
9 ft
105 feet
?
9 = 6105 ?
9 * ? = 105 * 6
9 * ? = 630
? = 70 feet
The silo is 70 feet tall.
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Find the perimeter and the area of a Trapezoid
2 m
3 m
4 m
A = (1/2)h(b1 + b2)
A = (1/2)(2)(3 + 4)
A = (1/2)(2)(7)
A = 1(7)
A = 7 square meters
5m5mPerimeter = 3m + 5m + 4m + 5m =
17m
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Circleradius is in greendiameter is in blue
2r = d Twice the radius is the diameter
CircumferenceC = 2r or 2r
AreaA = r2
Find the Circumference and the Area if the diameter is 22 in.
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Examples
Page 263 #8
V = Bh
V = (6 sq yd)*(6 yard)
V = 36 cubic yards
Page 263 #14
V = (1/3)Bh
V = (1/3)(78.5 sq ft)(24 ft)
V = 628 cubic feet
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Surface Area
• Remember surface area is the sum of the areas of the surfaces of a three-dimensional figure.
• Take your time and calculate the area of each side.
• Look for sides that have the same area to lessen the number of calculations you have to perform.
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Examples
Page 263 #8
Area of the 2 Bases3 yd * 2 yd = 6 sq yd
Area of 2 sides2 yd * 6 yd = 12 sq yd
Area of other 2 sides3 yd * 6 yd = 18 sq yd
Surface area6 + 6 + 12 + 12 + 18 + 18
= 72 sq yd
Page 263 #14
Surface area of a coneSA = [pi]r2 + [pi]r*sqrt[r2 + h2]
SA = 3.14 * (5)2 + 3.14 * 5 * sqrt[52 + 242]
SA = 3.14 * 25 + 3.14 * 5 * sqrt[25 + 576]
SA = 78.5 + 15.7 sqrt[601]
SA = 78.5 +
SA = sq ft
Polygons# of Sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
20 Icosagon
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Sum of Interior Angles
2 * 180 = 360 degrees
3 * 180 = 540 degrees
4 * 180 = 720 degrees
4 - 2 = 2
5 - 2 = 3
6 - 2 = 4
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• The sum of the measures of the interior angles of a n-sided polygon is
• (n - 2)*180 degrees
What is the sum of the measures of the interior angles of a nonagon?
n = 9 (9-2) * 180 = 7 * 180 = 1260 degrees
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EVERYONE: How many sides does a polygon have if thesum of the interior angles is 900 degrees?
• (n - 2) * 180 = 900
• Divide both sides by 180• n - 2 = 5
• Add 2 to both sides• n = 7 The polygon has 7 sides.
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Prisms
Pyramids