Welcome to MM150 Unit 6 Seminar

Line AB AB Ray AB AB Line segment AB AB

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.

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.

Angle D Side Vertex Side A F

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

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

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

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

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 = 50 [ang]CBD = 50 degrees X – 10 = 40 [ang]ABC = 40 degrees D C B A

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

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

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

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

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 = 7 The polygon has 7 sides.

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

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

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)

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

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

Prisms Pyramids

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

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.)

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.

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 6 + 6 + 12 + 12 + 18 + 18 = 72 sq yd Page 263 #14 Surface area of a cone SA = [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 + 24.5 SA = 103 sq ft