DISCOVERING GEOMETRY Over 6000 years ago, geometry consisted primarily of practical rules for measuring land and for

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1 Name Period GEOMETRY Chapter One BASICS OF GEOMETRY Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. In this course, you will study many amazing patterns that were discovered by people throughout history and all around the world. DISCOVERING GEOMETRY Over 6000 years ago, geometry consisted primarily of practical rules for measuring land and for building. In fact, the term geometry means. The rules were found by observation and experimentation. Some early geometric methods were very accurate. For example, the sides of the Great Pyramid at Gizeh were accurate to 1 centimeter in 180 meters (1/18,000 cm). A more formal study of geometry began with the Greeks. About 600 B.C. a Greek mathematician named Thales made a number of geometric conjectures with logical arguments. He used abstract diagrams to explore possible geometric relationships. Follow the steps below to find one of the results that Thales discovered. 1. Mark a point on the semicircle. Connect the point to the ends of the semicircle with a ruler. 2. Choose another point on the semicircle and follow the same procedure. 3. Repeat one more time. 4. What did Thales notice about the angles formed in each case? ORGANIZING GEOMETRY Until the time of Euclid (300 B.C.), geometry was a collection of ideas. Euclid organized this information into a system. He began with a few basic postulates,, or statements that are accepted as true. Then, reasoning logically from his postulates, he was able to prove a number theorems, or statements that are proven to be true. He published a 13 volume work called The Elements. In his systematic approach, figures are constructed using only a compass and a straightedge. Since Euclid s time The Elements has been translated into more languages and published in more editions than any other book except the Bible. USING GEOMETRY Geometric ideas are applied in many ways. We are surrounded by objects with pleasing or useful shapes. We see arcs of circles in rainbows, hexagons in honeycombs, cubes in salt crystals, and spheres in soap bubbles. Architects have made use of a wide variety of geometric shapes. You will learn to recognize and describe patterns of your own. Sometimes, patterns allow you to make accurate predictions. As you embark on your study of geometry, you may wonder how successful you will be in your efforts. Because geometry is a logical system, it is necessary to spend time on a regular basis to master the basic ideas contained within it. A positive attitude and a willingness to try to do your best are also important factors in determining how you will do. Pythagoras was a Greek geometer who lived about 2500 years ago.. He wondered whether he could teach geometry even to a reluctant student. After finding such a student, Pythagoras agreed to pay him an obel for each theorem that he learned. Because the student was very poor, he worked diligently. After a time, however, the student realized that he had become more interested in geometry than in the money he was accumulating. In fact, he became so intrigued with his studies that he begged Pythagoras to go faster, now offering to pay him back an obel for each new theorem. Eventually, Pythagoras got all of his money back! (Harold R. Jacobs) I hope you will find geometry both enjoyable and rewarding. I look forward to this school year, for the memories we will make and for the knowledge that we will learn. 1

2 Section Patterns and Inductive Reasoning GOAL 1: Finding and Describing Patterns Ex. 1: Sketch the next figure in the pattern. Ex. 2: Sketch the next figure in the pattern. Ex. 3: Describe a pattern in the sequence of numbers. Predict the next two numbers. a. 2, 5, 8, 11,, b. 27, 9, 3, 1,, c. 2, 5, 11, 23,,. Describe GOAL 2: Using Inductive Reasoning Vocabulary: Vocabulary words are highlighted in yellow in this book. A conjecture. Inductive Reasoning. A counterexample. Much of the reasoning in geometry consists of three stages. 1. Look for a Pattern Look at several examples. Use diagrams and tables to help discover a pattern. 2. Make a Conjecture Use the examples to make a general conjecture. 3. Verify the Conjecture Use logical reasoning to verify that the conjecture is true in all cases. Ex. 5 Complete the conjecture based on the pattern you observe. 3 8 = = = = = 3162 The product of an odd number and an even number is. Ex. 6 Complete the conjecture. Conjecture: The sum of the first n odd positive integers is? To show that a conjecture is false, you only need to find one counterexample. Ex. 7 Show the conjecture is false by finding a counterexample. a. The difference of two whole numbers is a whole number. b. All odd numbers are prime. Not every conjecture is known to be true or false. These are called unproven or undecided. Ex. 8 (Example 5 from text) In the early 1700s a Prussian mathematician named Goldbach noticed that many even numbers greater than 2 can be written as the sum of two primes. Show this is true for even #s 20 to 30. 2

3 Section 1.2 Points, Lines, and Planes GOAL 1 Using Undefined Terms and Definitions A definition uses known words to describe a new work. In geometry, some words, such as point, line, and plane are undefined terms. Although these words are not formally defined, it is important to have general agreement about what each word means. A point. A line In this book, lines are always straight lines. A plane A few concepts in geometry must also be commonly understood without being defined. One such concept is the idea that a point lies on a line or a plane. Collinear points. Coplanar points. Ex. 1 Decide whether the statement is true or false. a. Point X lies on line m. b. X, Y, and Z are collinear. c. Point W lies on line m. d. X, Y, and Z are coplanar. e. Point V lies on line l. f. V, Y, and X are collinear. g. X, Y, and V are collinear. h. X, Y, and V are coplanar. Ex. 2 Name a point that is collinear with the given points. a. B and E b. C and H c. D and G d. A and C e. H and E f. G and B g. B and I h. B and C Ex. 3 Name a point that is coplanar with the given points. a. M, N, and R b. M, N, and O c. M, T, and Q d. Q, T, and R e. T, R, and S f. Q, S, and O g. O, P, and M h. O,S, and R Another undefined concept in geometry is the idea that a point on a line is between two other points on the line. You can use this idea to define other important terms in geometry. The line segment or segment AB ( AB ). 3

4 The ray AB ( AB ). Note that is the same as, and is the same as, but and are not the same. They have different initial points and extend in different directions. Ex. 4 Draw three noncollinear points, J, K, and L. Then draw, JK, KL, LJ If C is between A and B, then CA and CB are. Ex. 5 Draw two lines. Label points on the lines and name two pairs of opposite rays. Ex. 6 Complete the sentence. a. AB consists of the endpoints A and B and all points on the line AB that lie. b. PQ consists of the initial point P and all points on the line PQ that lie. c. Two rays or segments are collinear if they. d. MN and ML are opposite rays if. GOAL 2: Sketching intersections of Lines and Planes. Ex. 6 Sketch the figure described. a. Three points that are coplanar but not collinear. b. Three lines that intersect at a single point. c. Three lines that intersect at two points. d. Three lines that intersect at three points. e. Two planes that intersect. e. Two planes that do not intersect. 4

5 Section 1.3 Segments and Their Measures GOAL 1: Using Angle Postulates A postulate or axiom is a. A theorem. POSTULATE 1 Ruler Postulate The points on a line can be matched one to one with the real numbers. The real numbers that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. d = A - B or B - A AB is also called the length of AB. The absolute value of a number is the distance it is from 0. Ex, 1 Evaluate the following Ex. 2 Measure the length of the segment to the nearest millimeter. A B POSTUTLATE 2 Segment Addition Postulate If B is between A and C, then AB +BC = AC. If AB + BC = AC, then B is between A and C. A B C Ex. 3 Draw a sketch of the three collinear points. Then write the Segment Addition Postulate for the points. a. T is between M and N. b. J is between S and H. Ex. 4 In the diagram of collinear points, GK = 24, HJ = 10, and GH = HI = IJ. Find each length. a. HI b. IJ c. GH d. JK e. IG f. IK 5

6 Ex. 5 Suppose J is between H and K. Use the Segment Addition Postulate to solve for x. Then find the length of each segment. a. HJ = 2x + 4 b. HJ = 5x -- 3 JK = 3x + 3 JK = 8x -- 9 KH = 22 KH = 131 GOAL 2: Using the Distance Formula The Distance Formula is a. Ex. 6 Evaluate each expression. a b. 2 5 c The Distance Formula If A x 1, y 1 and x 2, y 2 B are points in a coordinate plane, then the distance between A and B is AB 2 2 x x y y1 Ex. 7 Find the distance between each pair of points. D(1, 3), E( 2, 4), F(0, 4) DE = DF = EF = Ex. 8 Review: Plot A(1, 4) and B( 3, 2) on the graph. Congruent segments are. There is a special symbol,, for indicating congruence. Lengths are equal Segments are congruent AB = AD AB AD Ex. 9 Find the lengths of the segments. Tell whether any of the segments have the same length. 6

7 Section 1.4 Angles and their measures GOAL: 1 Using Angle Postulates An angle. The rays are the of the angle. The initial point is the of the angle. Two angles are adjacent angles if The angle that has sides AB and AC is denoted by BAC, CAB, A. The point A is the vertex of the angle. Ex. 1 Name the vertex and sides of the angle. Write two names for the angle. Ex. 2 Name the angles in the figure. The measure of A is denoted by. The measure of an angle can be approximated with a protractor, using units called degrees ( ). Congruent angles. Measures are equal Angles are congruent m BAC mdef BAC DEF is equal to is congruent to POSTULATE 3 Protractor Postulate Consider a point A on one side of OB. The rays of the form OA can be matched one to one with the real numbers 0 to 180. The measure of AOBis equal to the absolute value of the difference between the real numbers for OA and OB. Ex. 3 Use a protractor to measure each angle to the nearest degree. a. b. A point is in the interior of an angle if it is. A point is in the exterior of an angle if it is. POSTULATE 4 Angle Addition Postulate If P is in the interior of RST, then m RSP mpst mrst. 7

8 Ex. 4 Use the Angle Addition Postulate to find the measure of the unknown angle. a. b. mdef mabc GOAL 2: Classifying Angles Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0 and less than or equal to 180. Ex. 5 State whether the angle appears to be acute, right, obtuse, or straight. Then estimate its measure. a. b. c. Ex. 6 In a coordinate plane, 1. Plot the points and sketch ABC. 2. Classify the angle. 3. Write the coordinates of a point that lies in the interior of the angle and 4. Write the coordinates of a point that lies in the exterior of the angle. a. A(2, 4) b. A( 2, 1) B( 1, 1) B(1, 4) C(4, 1) C(7, 2) Section 1.5 Segment and Angle Bisectors GOAL 1: Bisecting a Segment The midpoint of a segment. In this book, matching red congruence marks identify congruent segments in diagrams. A segment bisector is a. Ex. 1 Use a compass and a straightedge (ruler) to construct a segment bisector and midpoint of AB. (Page 34) This will be turned in today during class. Explain here how to construct a segment bisector. If you know the coordinates of the endpoints of a segment, you can calculate the coordinates of the midpoint. You simply take the mean, or average of the x-coordinates and the average of the y-coordinates. (Add them and take half.) This is known as the midpoint formula. 8 x x 2 y, y2

9 Ex. 2 Find the coordinates of the midpoint of a segment with the given endpoints. a. A( 2, 4) b. C(2, 4) c. E( 3, 2) B(4, 6) D(0, 8) F(7, 5) Ex. 3 Find the coordinateds of the other endpoint of the segment with the given endpoint and the midpoint M. a. A( 8, 1) b. B( 3, 5) M( 0, 3) M( 7, 4) GOAL 2: Bisecting an Angle An angle bisector is a. Ex 4 Draw an angle with a straight edge. Use construction tools to find the bisector of the angle (page 36). This will be turned in today. Explain here how you bisect an angle bisector. uuu r Ex. 5 PT is the angle bisector of RPS. Find the two angle measures not given in the diagram. a. b. c. uuu r Ex. 6 BT bisects ABC. Find the value of x. a. b. 9

10 Section 1.6 Angle Pair Relationships GOAL 1: Vertical and Angles and Linear Pairs Two angles are vertical angles if. Two adjacent angles are a linear pair if. Ex. 1 Use the figure at the right. a. Are 1 and 2 adjacent? b. Are 1 and 2 a linear pair? c. Are 3 and 4 a linear pair? d. Are 2 and 5 vertical angles? e. Are 1 and 4 vertical angles? f. Are 3 and 5 vertical angles? Two important facts are listed below. We will study them more in Chapter 2. Vertical angles are congruent. The sum of the measures of angles that form a linear pair is 180. Ex. 2 Decide whether the statement is always, sometimes, or never true. a. If m 1 = 40, then m 3 = 140 b. If m 4 = 130, then m 3 = 50. c. 1 and 3 are congruent. d. m 1 + m 3 = m 2 + m 4. e. m 2 = 180 m3. Ex. 3 Use the figure at the right. a. If m 6 = 78, then m 7 =. b. If m 8 = 94, then m 6 =. c. If m 9 = 124, then m 8 =. d. If m 7 = 47, then m 9 =. e. If m 8 = 158, then m 9 =. f. If m 7 = 15, then m 6 =. Ex. 4 Find the value of the variable. a. 10

11 GOAL 2: Complementary and Supplementary Angles Two angles are complementary angles if. Each angle is the complement of the other. Complementary angles can be adjacent or nonadjacent. Two angles are supplementary angles if. Each angle is a supplement of the other. Supplementary angles can be adjacent or nonadjacent. Ex. 5 State whether the angles are complementary, supplementary, or neither. a. b. c. Ex. 6 Assume A and B are complementary and B and C are supplementary. a. If m A = 42, then m B = and m C =. b. If m B = 78, then m A = and m C =. c. If m A = 17, then m B = and m C =. d. If m B = 45, then m A = and m C =. Ex. 7 A and B are complementary. Find the m A and the m B. a. b. m A 5x 8 m A 3x 7 m B x 4 m B 11x 1 Ex. 8 A and B are supplementary. Find the m A and the m B. a. b. m A 3x m A 6x 1 m B x 8 m B 5x 17 Ex. 9 Find the values of the variables. 11

12 Section 1.7 GOAL 1: Reviewing Perimeter, Circumference, and Area In this lesson, you will review some common formulas for perimeter, circumference, and area. You will learn more about area in Chapters 6, 11, and 12. Ex. 1 Find the perimeter (or circumference) and area of the figure. Ex. 2 Find the area of the figure described. a. Rectangle with length 8 centimeters and width 4.5 centimeters. b. Triangle with height 5 inches and base 12 inches. c. Circle with diameter 10 feet. 12