1. Welcome to MM150 Unit 6
Seminar

2. Line AB AB
Ray AB AB
Line segment AB AB

3. Plane
Any three points that do not lie on the same line determine a plane. (Since 2 points determine a line, a line and a point not on the line determine a unique plane).
2. A line in a plane divides the plane into 3 parts: the line and 2 half-planes.
3. The intersection of 2 planes is a line.

4. 3 Definitions Parallel planes – 2 planes that do not intersect
Parallel lines – 2 lines IN THE SAME PLANE that do not intersect
Skew lines – 2 lines NOT IN THE SAME PLANE that do not intersect.

5. Angle D
Side
Vertex
Side A F

6. Angle Measures Acute Angle 0 degrees < acute < 90 degrees
Right Angle 90 degrees
Obtuse Angle 90 degrees < obtuse < 180 degrees
Straight Angle 180 degrees

7. More Angle Definitions
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 L M H B D

8. If the measure of [ang]LDM is 33 degrees, find the measures of the other 2 angles.
Given information:
[ang]BDH is a straight angle
[ang]BDM is a right angle L M H B D

9. If the measure of [ang]LDM is 33 degrees, find the measures of the other 2 angles.
Given information:
[ang]BDH is a straight angle
[ang]BDM is a right angle
[ang]BDM=90
[ang]BDL=90-33=57 deg
[ang]MDH=90 deg L M H B D

10. If [ang]ABC and [ang]CBD are complementary and [ang]ABC is 10 degrees less than [ang]CBD, find the measure of both angles.
[ang]ABC + [ang]CBD = 90
Let x = [ang]CBD
Then x – 10 = [ang]ABC
X + (x – 10) = 90
2x – 10 = 90
2x = 100
X = [ang]CBD = 50 degrees
X – 10 = [ang]ABC = 40 degrees D C B A

11. Polygons # of Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6
Hexagon 7
Heptagon 8
Octagon 9
Nonagon 10
Decagon 12
Dodecagon 20
Icosagon

12. 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

13. Sum of Interior Angles 2 * 180 = 360 degrees 4 - 2 = 2
5 - 2 = 3
4 * 180 = 720 degrees
6 - 2 = 4

14. EVERYONE: How many sides does a polygon have if the sum of the interior angles is 900 degrees?
Formula: (n - 2)*180 degrees, where n is number of sides of polygon

15. n = 7 The polygon has 7 sides.
EVERYONE: How many sides does a polygon have if the sum 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 = The polygon has 7 sides.

16. Similar Figures Y B 80[deg] 80[deg] 4 4 2 2 A 1 X 2 Z C 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 = 2
AB 2
YZ = 4 = 2
BC 2
XZ = 2 = 2
AC 1

17. 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?
9 = 6 ?
9 * ? = 105 * 6
9 * ? = 630
? = 70 feet
The silo is 70 feet tall. ?
6 ft
9 ft
105 feet

18. Units in measurement
Let’s consider a rectangle with length 5 inches and width 3 inches:
Perimeter = 2l + 2w = 10 in + 6 in = 16 inches
Area = l * w = 5 in * 3 in = 15 in*in = 15 in^2 (or 15 sq. in.)
Rectangular box with height 2 inches:
Volume = l * w * h = 5 in * 3 in * 2 in = 30 in*in*in = 30 in^3 (30 cubic inches)

19. Area of a Trapezoid 3 m 2 m 4 m A = (1/2)h(b1 + b2)
A = 7 square meters

20. Circle radius is in green diameter is in blue
2r = d Twice the radius is the diameter
Circumference
C = 2∏r or 2r∏
Since 2r = d
C = ∏d
Area
A = ∏r2

21. Prisms
Pyramids

22. Volume
In 3 dimensions, so general rule is that volume is base (area) times height (length)
For prisms V=Bh
For pyramids V=(1/3)Bh
Similarly with cylinders and cones
Page 255
Spheres
V = (4/3) pi * r^3
SA = 4 pi * r^2

23. Examples Page 263 #8 Rectangular prism (box) V = Bh
V = (6 sq yd)*(6 yard)
V = 36 cubic yards
Page 263 #14
cone
V = (1/3)Bh
V = (1/3)(78.5 sq ft)(24 ft)
V = 628 cubic feet
Page 263 #16
sphere
V = (4/3)pi*r^3
V = (4/3)(3.14)(7 mi)^3
V = 1436 cubic miles (approx.)

24. 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.

25. Examples of surface area
Page 263 #8
Area of the 2 Bases
3 yd * 2 yd = 6 sq yd
Area of 2 sides
2 yd * 6 yd = 12 sq yd
Area of other 2 sides
3 yd * 6 yd = 18 sq yd
Surface area
= 72 sq yd
Page 263 #14
Surface area of a cone
SA = [pi]r2 + [pi]r*sqrt[r2 + h2]
SA = 3.14 * (5) * 5 * sqrt[ ]
SA = 3.14 * * 5 * sqrt[ ]
SA = sqrt[601]
SA =
SA = sq ft