1. Points, Lines, Planes, and Angles

2.  A point, line, and plane are three basic terms in geometry that are NOT given a formal definition, yet we recognize them when we see them.  A line is a set of points.  Any two distinct points determine a line.

3.  Any point on a line separates the line into three parts: the point and two half lines.  A ray is a half line including the endpoint.  A line segment is part of a line between two points, including the endpoints.

4. Line segment AB Ray BA Ray AB Line AB SymbolDiagramDescription A B A A A B B B

5.  We can think of a plane as a two-dimensional surface that extends infinitely in both directions.  Any three points that are not on the same line (noncollinear points) determine a plane.  A line in a plane divides the plane into three parts, the line and two half planes.  Any line and a point not on the line determine a unique plane.  The intersection of two planes is a line.

6.  An angle is the union of two rays with a common endpoint; denoted.  The vertex is the point common to both rays.  The sides are the rays that make the angle.  There are several ways to name an angle: ABC, CBA, B

7.  The measure of an angle is the amount of rotation from its initial to its terminal side.  Angles can be measured in degrees, radians, or, gradients.  Angles are classified by their degree measurement.  Right Angle is 90°  Acute Angle is less than 90°  Obtuse Angle is greater than 90° but less than 180°  Straight Angle is 180°

8.  Adjacent Angles-angles that have a common vertex and a common side but no common interior points.  Complementary Angles-two angles whose sum of their measures is 90°.  Supplementary Angles-two angles whose sum of their measures is 180°.

9.  If ABC and CBD are supplementary and the measure of ABC is 110°. Determine the measure of CBD. A B C D 110°

10.  If ABC and CBD are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle. A B C D

11. A B C D ABC = 154.2° CBD = 25.7°

12.  Vertical angles are the nonadjacent angles formed by two intersecting straight lines.  Vertical angles have the same measure.  A line that intersects two different lines, at two different points is called a transversal.

13. 5 6 12 4 87 3 One interior and one exterior angle on the same side of the transversal–have the same measure Corresponding angles Exterior angles on the opposite sides of the transversal–have the same measure Alternate exterior angles Interior angles on the opposite side of the transversal–have the same measure Alternate interior angles 5 6 12 4 87 3 5 6 12 4 87 3

14. 120°

15. Polygons

16.  Polygons are named according to their number of sides (line segments). Icosagon20Heptagon7 Dodecagon12Hexagon6 Decagon10Pentagon5 Nonagon9Quadrilateral4 Octagon8Triangle3 NameNumber of Sides NameNumber of Sides

17.  The sum of the measures of the interior angles of a triangle is 180°. 30° 80° ____°

19. Acute Triangle All angles are acute. Obtuse Triangle One angle is obtuse.

20. Right Triangle One angle is a right angle. Isosceles Triangle Two equal sides. Two equal angles.

21. Equilateral Triangle Three equal sides. Three equal angles (60º) each. Scalene Triangle No two sides are equal in length.

22.  Two polygons are similar (~) if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion. 4 3 4 6 88 12 6

23. 8 3 6 5 y x

24.  Catherine Johnson wants to measure the height of a lighthouse. Catherine is 5 feet tall and determines that when her shadow is 12 feet long, the shadow of the lighthouse is 75 feet long. How tall is the lighthouse? ?

25. x 75 12 5

26.  If corresponding sides of two similar figures are the same length, the figures are congruent.  Corresponding angles of congruent figures have the same measure.

27.  Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360°.  Quadrilaterals may be classified according to their characteristics.

28.  Trapezoid Two sides are parallel.  Parallelogram Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.

29.  Rhombus Both pairs of opposite sides are parallel. The four sides are equal in length.  Rectangle Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.

30.  Square Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.

31. Perimeter and Area

32. P = s 1 + s 2 + b 1 + b 2 P = s 1 + s 2 + s 3 P = 2b + 2w P = 4s P = 2l + 2w Perimeter Trapezoid Triangle A = bhParallelogram A = s 2 Square A = lwRectangle AreaFigure

34.  Find the perimeter and area of the trapezoid. 16 in 6 in 13 in 12 in

35.  Circles  Circumference is the length around the outside of the circle.  Radius is a segment from the center of the circle to any point on the edge.  Diameter is a segment from edge to edge through the center. (Twice the radius!)  Formulas for area and circumference involve the constant pi or π.

36.  Circles  What is π?  We can just use the approximate value of 3.14, but…

37.  π = 3.14 159265 35897932384626433832795028841971 693993751058209749445923078164062862089986280 348253421170679821480865132823066470938446095 505822317253594081284811174502841027019385211 055596446229489549303819644288109756659334461 284756482337867831652712019091456485669234603 486104543266482133936072602491412737245870066 063155881748815209209628292540917153643678925 903 60011330530548820466521384146951941511609433057270 36575959195309 21861173819326117931051185480744623799627495 673518857527248912279381830119491298336733624406566430860213 949463952247371907021798609437027705392171762931767523846748 18467669405132000568127145263560 8277857713427577896091736371787 2146844090122495343014654958537105079227968925892354201995611212902 1960864034418159813629774771309960518707211349999998372978049951059 7317328160963185950244594553469083026425223082533446850352619311881 71 01000313783875288658753320838142061717766914730359825349042875546873115 9562 86388235378759375195778185778053217122680661300192787661119590921642019893809525720106548 58632788659361533818279682303019520353018529689957736225994138912497217752834791315155748 572424541506959508295331168617278558890750983817546374649393192550604009 277016711390098488…

38.  Terri is installing a new circular swimming pool in her backyard. The pool has a diameter of 27 feet. How much area will the pool take up in her yard? (Use π = 3.14.)

39. Volume and Surface Area

40.  Volume is the measure of the capacity of a figure. It is the amount of material you can put inside a three-dimensional figure.  Surface area is sum of the areas of the surfaces of a three-dimensional figure. It refers to the total area that is on the outside surface of the figure.

41. Volume – the amount of physical space the object occupies Surface Area – unfolding and finding the area of each face 3 units 5 units2 units

42. Sphere Cone V =  r 2 h Cylinder V = s 3 Cube V = lwhRectangular Solid DiagramFormulaFigure

43. Sphere Cone SA = 2  rh + 2  r 2 Cylinder SA= 6s 2 Cube SA = 2lw + 2wh +2lhRectangular Solid DiagramFormulaFigure

45.  Mr. Stoller needs to order potting soil for his horticulture class. The class is going to plant seeds in rectangular planters that are 12 inches long, 8 inches wide and 3 inches deep. If the class is going to fill 500 planters, how many cubic inches of soil are needed? How many cubic feet is this?

46.  We need to find the volume of one planter.  Soil for 500 planters would be

47.  where B is the area of the base and h is the height.  Example: Find the volume of the pyramid. 12 m 18 m