Points, Lines, Planes, and Angles 6.1 Points, Lines, Planes, and Angles

Basic Terms 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 unique line. 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.

Basic Terms Line segment AB Ray BA Ray AB Line AB Symbol Diagram Description A B

Plane 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 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.

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:

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° Acute Angle is less than 90° Obtuse Angle is greater than 90° but less than 180° Straight Angle is 180°

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.

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

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

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.

Special Names Alternate interior angles Alternate exterior angles 5 6 1 2 4 8 7 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

6.2 Polygons

Polygons Polygons are named according to their number of sides. Icosagon 20 Heptagon 7 Dodecagon 12 Hexagon 6 Decagon 10 Pentagon 5 Nonagon 9 Quadrilateral 4 Octagon 8 Triangle 3 Name Number of Sides

Types of Triangles Acute Triangle All angles are acute. Obtuse Triangle One angle is obtuse.

Types of Triangles continued Right Triangle One angle is a right angle. Isosceles Triangle Two equal sides. Two equal angles.

Types of Triangles continued 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. 4 3 6 9 4.5

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 75 12 5

Example continued Therefore, the lighthouse is 31.25 feet tall. x 5 12 75 12 5 Therefore, the lighthouse is 31.25 feet tall.

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.

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

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

Classifications continued 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.

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

6.3 Perimeter and Area

Formulas Figure Perimeter Area Rectangle P = 2l + 2w A = lw Square P = 4s A = s2 Parallelogram P = 2b + 2w A = bh Triangle P = s1 + s2 + s3 Trapezoid P = s1 + s2 + b1 + b2

Example Marcus Sanderson needs to put a new roof on his barn. One square of roofing covers 100 ft2 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.

Example continued a) The area of the roof is A = 30(50) A = 1500 ft2 A = lw A = 30(50) A = 1500 ft2 1500 x 2 (both sides of the roof) = 3000 ft2 b) Determine the number of squares

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

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. leg2 + leg2 = hypotenuse2 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 a2 + b2 = c2 a b c

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

Example continued The distance is approximately 36.06 feet. 12 38 b

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.

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.

Volume and Surface Area 6.4 Volume and Surface Area

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 the 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.

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?

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

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

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.

Volume of a Prism V = Bh, where B is the area of the base 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