MATH STUDENT BOOK. 10th Grade Unit 3

Transcription

1 MTH STUENT K 10th Grade Unit 3

2 Unit 3 ngles and Parallels MTH 1003 NGLES N PRLLELS INTRUTIN 3 1. NGLE EFINITINS N MESUREMENT 5 NGLE EFINITINS 5 NGLE MESUREMENT 12 SELF TEST NGLE RELTINSHIPS N THEREMS 21 RELTINSHIP EFINITINS 21 THEREMS 24 SELF TEST PRLLELS 38 SI PRPERTIES 38 TRNSVERSL N SPEIL NGLES 41 SELF TEST PPLYING PRLLELS T PLYGNS 58 TRINGLES 58 THER PLYGNS 70 SELF TEST 4 76 GLSSRY 80 LIFEP Test is located in the center of the booklet. Please remove before starting the unit. 1

3 ngles and Parallels Unit 3 uthor: Milton R. hristen, M.. Editor-in-hief: Richard W. Wheeler, M..Ed. onsulting Editor: Robert L. Zenor, M.., M.S. Revision Editor: lan hristopherson, M.S. Page 39: omstock, Stockbyte, Thinkstock 804 N. 2nd ve. E. Rock Rapids, I MMXVII by lpha mega Publications a division of Glynlyon, Inc. ll rights reserved. LIFEP is a registered trademark of lpha mega Publications, Inc. ll trademarks and/or service marks referenced in this material are the property of their respective owners. lpha mega Publications, Inc. makes no claim of ownership to any trademarks and/ or service marks other than their own and their affiliates, and makes no claim of affiliation to any companies whose trademarks may be listed in this material, other than their own. 2

4 Unit 3 ngles and Parallels ngles and Parallels Introduction In this LIFEP, we shall study another basic geometric idea. Lines, segments, and rays can be placed in such a way as to form angles. ngles are present everywhere and are very important in the study of geometry. Many angle relationships will be presented, along with methods for measuring angles. We shall also learn about parallels and how special angles are formed using parallels. Many theorems will be presented in connection with angles and parallels. bjectives Read these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEP. When you have finished this LIFEP, you should be able to: 1. Identify angles as acute, right, or obtuse. 2. Find the measure of angles with a protractor. 3. dd and subtract measures of angles. 4. Find the measure of angles by their relationship with other angles. 5. Prove theorems about angles. 6. efine terms related to parallels. 7. Prove theorems about parallel and related angles. 8. lassify triangles by their sides and by their angles. 9. Prove theorems about triangles and their related angles. 3

5 Unit 3 ngles and Parallels 1. NGLE EFINITINS N MESUREMENT To continue our study of geometry, we must learn basic angle definitions and measurement methods. efinitions will help in identifying and classifying angles; measurement will allow us to add and subtract angles. Section bjectives Review these objectives. When you have completed this section, you should be able to: 1. Identify angles as acute, right, or obtuse. 2. Find the measure of angles with a protractor. 3. dd and subtract measures of angles. NGLE EFINITINS These definitions relate to angles. Make sure you know them, because you will be using them many times in this LIFEP. If no confusion will result, the vertex letter can be used alone; or a numeral can be used. You should never use a single letter when several angles have the same vertex. EFINITIN ngle (+): the union of two noncollinear rays that have a common end point. Model: The two rays that form the angle are called its sides, and the common end point is called the vertex of the angle. The symbol for angle is +. Models: 1 The angle to the left is formed by the union of " and ". Its sides are " and ". Its vertex is. We can name the angle +, +, +, or +1. The angle to the right is the union of SR " and ST". The vertex is point S. This angle can be called+rst, +TSR, +S, or +2. R 2 S T E + could mean: " " +, the angle formed by " and " ; or +, the angle formed by " and " ; or +E, the angle formed by " and E " ; or +, the angle formed by " and " ; or +E, the angle formed by " and E " ; or +E, the angle formed by and E. Notice that when three letters are used to name an angle, the vertex letter is always placed between the other two. Section 1 5

6 ngles and Parallels Unit 3 EFINITIN cute angle: an angle whose measure is less than 90. Models: These angles are all acute angles. EFINITIN Right angle (rt. +): an angle whose measure equals 90. Models: These angles are all right angles. little box at the vertex indicates a right angle. EFINITIN btuse angle: an angle with a measure greater than 90 but less than 180. Models: These angles are all obtuse angles. EFINITIN isector of an angle: a ray that is in the interior of the angle and divides the angle into two angles of equal measure. Model: If the measure of + (m +) equals the measure of + (m +), then " bisects +. 6 Section 1

7 Unit 3 ngles and Parallels Given the following angle, complete the following items. R S 1 T 1.1_ 1.2_ 1.3_ State four ways of naming this angle. What point is the vertex of this angle? Name the sides of this angle. Write your answers on the lines. 1.4_ 1.5_ Name the sides of +. Name the vertex of +. Given the following angles, complete the following items. F E 1.6_ 1.7_ 1.8_ 1.9_ Name three angles with vertex F. " " F and FE are the sides of what angle? What ray is the common side of +F and+fe? If m +FE = m +F, name the bisector of +FE. Section 1 7

8 ngles and Parallels Unit 3 Given the following angles, complete the following items. 1.10_ Name the acute angles. 1.11_ Name the obtuse angles. omplete the following activity. 1.12_ In the space provided: a. raw a right angle. b. raw a bisector of the right angle. Label all parts. c. What kind of angles are formed by the bisector? 8 Section 1

9 Unit 3 ngles and Parallels EFINITIN Perpendicular lines: two lines that intersect such that the four angles formed are equal to each other. l Model: m When m +1 = m +2 = m +3 = m +4, we say that line l is perpendicular (9) to line m. Segments and rays that are parts of lines can also be called perpendicular if the lines of which they are part are perpendicular. Models: T R S U " 9 " RS 9 TU 9 etweenness of rays means that for any +, " is between " and " when all three rays have the same end point and when lies in the interior of +. oth of these conditions must be true for " to be between " and ". Section 1 9

10 ngles and Parallels Unit 3 Models N Q M P E (INTERIR) " " " MQ is between MN and MP. " " " is between and E. " " " is not between and. It is not in the interior of +. omplete the following activities. 1.13_ a. raw lines 9 to l through points,,, and. l _ b. What does inductive reasoning tell you about these lines? 1.14_ a. raw ray " between ray " and ". _ b. What does inductive reasoning tell you about the sum of the measures of + and +? 10 Section 1

11 Unit 3 ngles and Parallels 1.15_ a. raw RS " 9 TU " at. raw " between T" and S ". raw " opposite ". b. What does inductive reasoning tell you about m +S and m +R? c. bout m +T and m +U? 1.16_ a. raw several 9 lines. _ b. What kinds of angles are formed? (acute, obtuse, or right) 1.17_ a. Using ", ", ", and ", make a diagram so that + is an acute +, + is an obtuse +, and + is a right +. _ b. What kind of angle is +? 1.18_ What is the measure of a rt. +? 1.19_ What is the measure of an obtuse +? 1.20_ What is the measure of an acute +? Section 1 11

12 ngles and Parallels Unit 3 NGLE MESUREMENT We now need to work with angle measurement. The following postulates will help us to establish a method for measuring angles. P6 tells us that every angle has a measure that is a real number between 0 and 180. No angle has a measure of 0 ; no angle has a measure of 180. The next postulate tells us how to find the measurement number. PSTULTE 6 P6: Every angle corresponds with a unique real number greater than 0 and less than 180. (angle measurement postulate) PSTULTE 7 P7: The set of rays on the same side of a line with a common end point in the line can be put in one-to-one correspondence with the real numbers from 0 to 180 inclusive in such a way: 1. that one of the two opposite rays lying in the line is paired with zero and the other is paired with that the measure of any angle whose sides are rays of that given set is equal to the absolute value of the difference between the numbers corresponding to its sides. (protractor postulate) P7 tells us how to build and use a protractor to measure an angle. Take a line and a point on that line. Place a set of rays all on the same side of the line with common end point. Then match the numbers from 0 to 180 inclusive with the rays in such a way that one of the two opposite rays is paired with zero and the other is paired with Section 1

13 Unit 3 ngles and Parallels You have built a protractor. Now, to find the measure of an angle, place the sides of the angle along the rays of the protractor, lining up the vertex of the angle with point. Noting the numbers associated with the rays, subtract one ray number from another and take the absolute value of the difference to make the answer a positive number. This number is the measure of the angle. You can line up one ray of the angle with the zero ray, but this step is not necessary. You only need to have the vertex matching the mark on the protractor. G F E m+ = 20 0 = 20 = 20 m+ = = 20 = 20 m+ = = 50 = 50 The order in which you subtract does not affect the measure. m+p = = -50 = 50 m+f = = -110 = 110 m+g = = -90 = 90 Refer to this diagram to complete the following items. E Z X 1.21_ Write the measure of the following + s. a. m+x = b. m+x = c. m+ = d. m+ = e. m+x = f. m+e = g. m+e = h. m+ = i. m+ = Section 1 13

14 ngles and Parallels Unit 3 SELF TEST 1 omplete the following items (each answer, 3 points). 1.01_ raw an acute angle and label it so its name is +WN. 1.02_ Name the vertex of +WN. 1.03_ Name " the sides of +WN. 1.04_raw S the bisector of +T. T 1.05_raw ) R 9 ) TU at point Q. T Q U 1.06_ 1.07_ What is the measure of an obtuse angle? What is the measure of an acute angle? 18 Section 1

15 Unit 3 ngles and Parallels Given the following diagram, write the required information (each answer, 3 points). X 1.08 m +X = m +X = m +X = 1.09 m +X = m +X = m +X = 1.010_ Name the largest angle in the figure _m +X + m +X = Write the required information (each answer, 3 points) _ an we have an angle with a measure of 0 degrees? 1.013_ an we have an angle with a measure of 180 degrees? Find the sum Find the difference Section 1 19

16 ngles and Parallels Unit 3 Supply the reasons in the following proof (each answer, 4 points). R S T P W Given: To Prove: m +RPS = m +TPW m +RPT = m +SPW STTEMENTT 1. m +RPS = m +TPW 2. m +RPS + m +SPT = m +TPW + m +SPT 3. m +RPS + m +SPT = m +RPT 4. m +TPW + m +SPT = m +SPW 5. m +RPT = m +SPW RESN SRE TEHER initials date 20 Section 1

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