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Chapter 6 Geometry 3

WHAT WE WILL COVER Points, lines, planes, and angles Polygons, similar figures, and congruent figures Perimeter and Area Pythagorean theorem Circles Volume and Surface Area 4

6.1 Points, Lines, Planes, and Angles Page 216 5

Basic Terms A point has no dimension. A line, a set of points, has 1 dimension. A line can be determined by any two distinct points. 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. 6

Basic Terms Line segment AB Ray BA Ray AB Line AB SymbolDiagramDescription A B A A A B B B 7

Plane A plane is a two-dimensional surface that extends infinitely in both directions. Any three points that are not on the same line determine a unique 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. Page 220 8

Angles 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: 9

Angles 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 o –Acute Angle is less than 90 o –Obtuse Angle is greater than 90 o but less than 180 o –Straight Angle is 180 o 10

Some Types of Angles 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 degrees. Supplementary Angles-two angles whose sum of their measures is 180 degrees. Page

Example If are supplementary and the measure of is 6 times larger than, determine the measure of each angle. A B C D 12

More definitions 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. Special angles are given to the angles formed by a transversal crossing two parallel lines. 13

Special Names 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 Page

Find the measure of the other angles Example 94 o 15

6.2 Polygons Page

Polygons Polygons are named according to their number of sides. Icosagon20Heptagon7 Dodecagon12Hexagon6 Decagon10Pentagon5 Nonagon9Quadrilateral4 Octagon8Triangle3 NameNumber of Sides NameNumber of Sides 17

The sum of the measures of the interior angles of an n-sided polygon is (n - 2)180 o. Example: A certain brick paver is in the shape of a regular octagon. Determine the measure of an interior angle and the measure of one exterior angle. Page

Determine the sum of the interior angles. The measure of one interior angle is The exterior angle is supplementary to the interior angle, so 180 o o = 45 o 19

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

Similar Figures Two polygons are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion

Example 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? x

Example continued x Therefore, the lighthouse is feet tall. 23

Congruent Figures If corresponding sides of two similar figures are the same length, the figures are congruent. Corresponding angles of congruent figures have the same measure. Page

Quadrilaterals Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360 o. Quadrilaterals may be classified according to their characteristics. Page 234 Trapezoid Parallelogram Rhombus Rectangle Square

6.3 Perimeter and Area Page

Formulas 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 Page

Example Marcus Sanderson needs to put a new roof on his barn. One square of roofing covers 100 ft 2 and costs $32.00 per square. If one side of the barn roof measures 50 feet by 30 feet, determine a) the area of the entire roof. b) how many squares of roofing he needs. c) the cost of putting on the roof. 28

Example continued a) The area of the roof is A = lw A = 30(50) A = 1500 ft x 2 (both sides of the roof) = 3000 ft 2 b) Determine the number of squares 29

Example continued c) Determine the cost 30 squares x $32 per square $960 It will cost a total of $960 to roof the barn. 30

Pythagorean Theorem The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse. leg 2 + leg 2 = hypotenuse 2 Symbolically, if a and b represent the lengths of the legs and c represents the length of the hypotenuse (the side opposite the right angle), then a 2 + b 2 = c 2 a b c Page

Example Tomas is bringing his boat into a dock that is 12 feet above the water level. If a 38 foot rope is attached to the dock on one side and to the boat on the other side, determine the horizontal distance from the dock to the boat. 12 ft 38 ft rope 32

Example continued The distance is approximately feet b 33

Circles A circle is a set of points equidistant from a fixed point called the center. A radius, r, of a circle is a line segment from the center of the circle to any point on the circle. A diameter, d, of a circle is a line segment through the center of the circle with both end points on the circle. The circumference is the length of the simple closed curve that forms the circle. Page

Example 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.) The radius of the pool is 13.5 ft. The pool will take up about 572 square feet. 35

6.4 Volume and Surface Area Page

Volume 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. 37

Volume Formulas Sphere Cone V =  r 2 h Cylinder V = s 3 Cube V = lwhRectangular Solid DiagramFormulaFigure Page

Surface Area Formulas Sphere Cone SA = 2  rh + 2  r 2 Cylinder SA= 6s 2 Cube SA = 2lw + 2wh +2lhRectangular Solid DiagramFormulaFigure 39

Example 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? 40

Example continued We need to find the volume of one planter. Soil for 500 planters would be 500(288) = 144,000 cubic inches 41

Polyhedron A polyhedron is a closed surface formed by the union of polygonal regions. Page

Euler’s Polyhedron Formula Number of vertices - number of edges + number of faces = 2 Example: A certain polyhedron has 12 edges and 6 faces. Determine the number of vertices on this polyhedron. # of vertices - # of edges + # of faces = 2 There are 8 vertices. 43

Volume of a Prism V = Bh, where B is the area of the base, look up the correct formula, and h is the height. Example: Find the volume of the figure. Area of one triangle. Find the volume. 8 m 6 m 4 m 44

Volume of a Pyramid where B is the area of the base, look up the correct formula, and h is the height. Example: Find the volume of the pyramid. Base area = 12 2 = m 18 m Page