Rays and Angles Examples

Transcription

1 Rays and Angles Examples 1. In geometry, an angle is defined in terms of two rays that form the angle. You can think of a ray as a segment that is extended indefinitely in one direction. Rays have exactly one endpoint, and that point is always named first when naming the ray. B C E F A D 2. Like segments, rays can also be defined using betweenness of points. Ray PQ, written PQ, consists of the points on PQ and all points S on PQ such that Q is between P and S. 3. Any given point on a line determines exactly two rays called opposite rays. This point is the common endpoint of the opposite rays. In the figure below, PQ and PR are opposite rays, and P is the common endpoint. 4. Opposite rays can be defined as a figure formed by two collinear rays with a common endpoint, since the two rays lie on the same line.

2 5. Similarly, an angle can be defined as a figure formed by two rays with a common endpoint. The two rays are called the sides of the angle. The common endpoint is called the vertex. 6. The figure formed by opposite rays is often referred to as a straight angle. Straight angles have a degree measure of 180 degrees. 7. In the figure at the right, the sides of the angle are YX and YZ, and the vertex is Y. This angle could be named Y, XYZ, ZYX, or 1. When letters are used to name an angle, the letter that names the vertex is used either as the only letter or as the middle of three letters. 8. A single letter names an angle only when there is no chance of confusion. For example, it is not obvious which angle shown at the right is A since there are three different angles that have A as a vertex. Name the three angles. BAD, BAC, and CAD 9. Example Whenever two or more angles have a common vertex, you need to use either three letters or a number to name each angle. Refer to the figure at the right to answer each question. a. What number names QSP? 3 b. What is the vertex of 2? Q c. What are the sides of 1? SQ and SR

3 10. Just as a ruler can be used to measure the length of segment, a protractor can be used to find the measure of an angle in degrees. To find the measure of an angle, place the center of the protractor over the vertex of the angle. Then align the mark labeled 0 on either side of the scale with one side of the angle. This has been done for XYZ shown below. Z Y X Using the inner scale of the protractor, shown in red, you can see that Y is a 40- degree (40 o ) angle. Thus, we say that the degree measure of XYZ is 40. This can also be written as m XYZ = Protractor Postulate Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on each side of AB, such that the measure of the angle formed is r. The protractor postulate guarantees that there is only one 40 o angle on each side of YX. 12. Example Use a protractor to find the degree measure of each number angle. m 1 = 30 m 2 = 95 m 3 = 18 m 4 = 37

4 13. In the figure below, you can see that point R is in the interior of PQS, m PQS = 110 and m RQS = 30. The sides of PQR align with the marks labeled 110 and 30 on the inner scale. So m PQR = or 80. Since = 110, m PQR + m RQS = m PQS. This example and others like it, lead us to the Angle Addition Postulate. P R Q 14. Angle Addition Postulate If R is in the interior of PQS, then m PQR + m RQS = m PQS. If m PQR + m RQS = m PQS, then R is in the interior of PQS. S 15. Example Captain Julie Wright, a pilot for United Airlines, is on approach for a landing at Chicago s O Hare International Airport. Her present compass heading is 73 degrees. This heading refers to the measurement of the angle formed by the flight path of the plane and an imaginary path in the direction due north. The tower has informed Captain Wright to land on runway 9. She knows that by multiplying the runway number by 10 degrees will give her the compass heading for a landing on that runway. So the compass heading for her landing must be 90 degrees. How many degrees and in what direction must Captain Wright turn in order to land on runway 9?

5 Let AP represent the path of Captain Wright s plane, and let AR represent the path for a landing on runway 9. Determine the number of degrees that the plane must be turned to land on runway 9 by determining the measure of PAR. The compass heading for the path of Captain Wright s plane, AB, is 73 o. Using the formula given in the problem, we know that the compass heading for a landing on runway 9 is 9(10) or 90 o. AP and AR represent the paths corresponding to these compass headings. We can use the angle addition postulate to find m PAR. m NAP + m PAR = m NAR 73 + m PAR = 90 m PAR = 17 Thus, Captain Wright must turn the plane 17 o right to land on runway Thought Provoker Suppose that Captain Wright s plane was told to land on runway 6 instead of runway 9. The compass heading for her plane is still 73 o. Determine the number of degrees that the plane must be turned to land on Runway 6. m NAR + m PAR = m NAP 60 + m PAR = 73 m PAR = 13 Captain Wright must turn the plane 13 o left to land on Runway 6. Runway 6 Path of plane

6 Name: Date: Class: Rays and Angles Activity Sheet 1. What are two other names for QS? 2. What is the endpoint of SP? 3. True or false: RX and RT are opposite rays? Why? 4. What are the sides of 2? 5. Name all of the angles that have RY for a side. 6. Complete: m XRT = m 2 +. A B C D E P Q R Find the measure of the following angles from above. 7. PQA 8. RQE 9. PQC 10. AQB 11. BQD 12. EQC 13. AQC 14. AQE

7 4 Use the figure above to answer each question 15. What is the vertex of angle 2? 16. Name a straight angle. 17. Name all the angles that have J as the vertex. 18. Do 3 and 4 have a common side? If so, name it. 19. Do 2 and J name the same angle? Explain. In the figure, XP and XT are opposite rays. Given the following conditions, find the value of x and the measure of the indicated angle. 20. m SXT = 3x 4, m RXS = 2x + 5, m RXT = 111, find m RXS. 21. m PXQ = 2x, m QXT = 5x 23, find m QXT. 22. m QXR = x + 10, m QXS = 4x 1, m RXS = 91, find m QXS. 23. m QXR = 3x + 5, m QXP = 2x 3, m RXP = x + 50, find m RXT.

8 Name: Date: Class: Rays and Angles Activity Sheet Key 1. What are two other names for QS? QR, QT 2. What is the endpoint of SP? S 3. True or false: RX and RT are opposite rays? Why? No, the rays do not lie on the same line. 4. What are the sides of 2? RX, RY 5. Name all of the angles that have RY for a side. PRY, XRY, TRY 6. Complete: m XRT = m 2 +. m 3 A B C D E R P Q Find the measure of the following angles from above. (Answers are approximations.) 7. PQA 10 o 8. RQE 25 o 9. PQC 105 o 10. AQB 30 o 11. BQD 80 o 12. EQC 50 o 13. AQC 95 o 14. AQE 145 o

9 4 Use the figure above to answer each question. 15. What is the vertex of angle 2? J 16. Name a straight angle. HUK 17. Name all the angles that have J as the vertex. HJK, HJU, UJK 18. Do 3 and 4 have a common side? If so, name it. No 19. Do 2 and J name the same angle? Explain. No, since J could refer to 1, 2, or HJK. In the figure, XP and XT are opposite rays. Given the following conditions, find the value of x and the measure of the indicated angle. 20. m SXT = 3x 4, m RXS = 2x + 5, m RXT = 111, find m RXS. SXT + RXS = RXT (3x 4) + (2x + 5) = 111 5x + 1 = 111 5x = 110 x = 22 RXS = 2(22) o

10 21. m PXQ = 2x, m QXT = 5x 23, find m QXT. m PXQ + QXT = 180 (2x) + (5x 23) = 180 7x 23 = 180 7x = 203 x = 29 m QXT = 5(29) 23 = 122 o 22. m QXR = x + 10, m QXS = 4x 1, m RXS = 91, find m QXS. m QXR + RXS = QXS (x + 10) + (91) = (4x 1) x = 4x = 3x x = 34 QXS = 4(34) 1 = 135 o 23. m QXR = 3x + 5, m QXP = 2x 3, m RXP = x + 50, find m RXT. m QXR + m QXP = m RXP (3x + 5) + (2x 3) = (x + 50) 5x + 2 = x x = 48 x = 12 m RXT + m RXP = 180 m RXT + ( ) = 180 m RXT = 118 o

11 Student Name: Date: Rays and Angles Checklist (This is a suggested checklist if you are using this as a number grade; on the other hand, you could devise your own rubric.) 1. On question 1, did the student give two other names for QS? b. Student gave one correct name (5 points) 2. On question 2, did the student give the correct endpoint? a. Yes (5 points) 3. On question 3, did the student answer (all) parts of the question correctly? b. Student answered correctly but did not describe why (5 points) 4. On question 4, did the student give correct sides? a. Both (10 points) b. One of two (5 points) 5. On question 5, did the student name all the angles? a. All three (15 points) b. Two of the three (10 points) c. One of the three (5 points) 6. On questions 6 thru 16, did the student answer questions correctly? a. All eleven (55 points) b. Ten of the eleven (50 points) c. Nine of the eleven (45 points) d. Eight of the eleven (40 points) e. Seven of the eleven (35 points) f. Six of the eleven (30 points) g. Five of the eleven (25 points) h. Four of the eleven (20 points) i. Three of the eleven (15 points) j. Two of the eleven (10 points) k. One of the eleven (5 points) 7. On question 17, did the student name all the angles? a. All three (15 points) b. Two of the three (10 points) c. One of the three (5 points) 8. On question 18, did the student answer question correctly? a. Yes (5 points)

12 9. On question 19, did the student answer (all) parts of the question correctly? b. Yes, but did not explain 10. On question 20, did the student find the value of x and the measure of the missing angle? b. Found the value of x but not the missing angle (5 points) 11. On question 21, did the student find the value of x and the measure of the missing angle? b. Found the value of x but not the missing angle (5 points) 12. On question 22, did the student find the value of x and the measure of the missing angle? b. Found the value of x but not the missing angle (5 points) 13. On question 23, did the student find the value of x and the measure of the missing angle? b. Found the value of x but not the missing angle (5 points) Total Number of Points A 162 points and above Any score below C needs remediation! B 144 points and above C 126 points and above D 108 points and above F 107 points and below