1. Welcome to MM150 Unit 6 Seminar

2. 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

3. Angle Two rays that come together at a vertex A D F Vertex Side

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

5. More Angle Definitions BD H LM 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

6. If the measure of [ang]LDM is 33 degrees, find the measures of the other 2 angles. BD H LM Given information: [ang]BDH is a straight angle [ang]BDM is a right angle

7. 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 C D [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

8. 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

9. 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 9

10. Example 1 2 3 4 5 6 7 8 If the measure of <1 is 45 degrees, find the remaining measures. 10

11. 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

12. 12 Similar Figures A B C X Y Z 80[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 = 2 AB 2 22 1 2 4 4 YZ = 4 = 2 BC 2 XZ = 2 = 2 AC 1

13. 13 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 = 6 105 ? 9 * ? = 105 * 6 9 * ? = 630 ? = 70 feet The silo is 70 feet tall.

14. 14 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 5m Perimeter = 3m + 5m + 4m + 5m = 17m

15. 15 Circle radius is in green diameter is in blue 2r = d Twice the radius is the diameter Circumference C = 2  r or 2r  Area A =  r 2 Find the Circumference and the Area if the diameter is 22 in.

16. 16 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

17. 17 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.

18. 18 Examples 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]r 2 + [pi]r*sqrt[r 2 + h 2 ] SA = 3.14 * (5) 2 + 3.14 * 5 * sqrt[5 2 + 24 2 ] SA = 3.14 * 25 + 3.14 * 5 * sqrt[25 + 576] SA = 78.5 + 15.7 sqrt[601] SA = 78.5 + SA = sq ft

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

20. 20 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

21. 21 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

22. 22 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.

23. 23 Prisms Pyramids