14.1 Central Angles and Inscribed Angles

Transcription

1 Name lass ate 14.1 entral ngles and Inscribed ngles ssential Question: How can you determine the measures of central angles and inscribed angles of a circle? Resource Locker xplore G.5. Investigate patterns to make conjectures about geometric relationships, including... special segments and angles of circles. lso G.12. Investigating entral ngles and Inscribed ngles chord is a segment whose endpoints lie on a circle. central angle is an angle less than 180 whose vertex lies at the center of a circle. n inscribed angle is an angle whose vertex lies on a circle and whose sides contain chords of the circle. The diagram shows two examples of an inscribed angle and the corresponding central angle. Houghton Mifflin Harcourt Publishing ompany hords entral ngle Inscribed ngle and Use a compass to draw a circle. Label the center. Use a straightedge to draw an acute inscribed angle on your circle from Step. Label the angle as. Use a straightedge to draw the corresponding central angle,. Module Lesson 1

2 Use a protractor to measure the inscribed angle and the central angle. Record the measure of the inscribed angle, the measure of the central angle, and the measure of 360 minus the central angle. List your results in the table. ngle Measure ircle ircle 2 ircle 3 ircle 4 ircle 5 ircle 6 ircle 7 m m m Repeat Steps - six more times. xamine a variety of inscribed angles (two more acute, one right, and three obtuse). Record your results in the table in Step. Reflect 1. xamine the values in the first and second rows of the table. Is there a mathematical relationship that exists for some or all of the values? Make a conjecture that summarizes your observation. 2. xamine the values in the first and third rows of the table. Is there a mathematical relationship that exists for some or all of the values? Make a conjecture that summarizes your observation. Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

3 xplain 1 Understanding rcs and rc Measure n arc is a continuous portion of a circle consisting of two points (called the endpoints of the arc) and all the points on the circle between them. rc Measure igure minor arc is an arc whose points are on or in the interior of a corresponding central angle. The measure of a minor arc is equal to the measure of the central angle. m = m major arc is an arc whose points are on or in the exterior of a corresponding central angle. The measure of a major arc is equal to 360 minus the measure of the central angle. m = 360 -m semicircle is an arc whose endpoints are the endpoints of a diameter. The measure of a semicircle is 180. djacent arcs are arcs of the same circle that intersect in exactly one point. and are adjacent arcs. Houghton Mifflin Harcourt Publishing ompany rc ddition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m = m + m Module Lesson 1

4 xample 1 If m = 18 and m = 33, determine m using the appropriate theorems and postulates. and intersect at point. If m = 33, then m = 33. If m = 33, then m = 33 by the Vertical ngles Theorem. If m = 33 and m = 18, then m = 33 and m = 18. y the rc ddition Postulate, m = m + m, and so m = 51. If m JK = 27, determine m NP using the appropriate theorems and postulates. MK and NJ intersect at point. If m JK = 27, then m JK = 27. If m JK = 27, then m = 27 by the. If m MN = 27 and m MP =, then m MN = 27 J M N K P and m MNP =. y the, m MNP = m MN + m NP, and so m NP = m - m MN = Reflect 3. The minute hand of a clock sweeps out an arc as time moves forward. rom 3:10 p.m. to 3:30 p.m., what is the measure of this arc? xplain your reasoning : : Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

5 Your Turn 4. If m = 45 and m = 56, determine m using the appropriate theorems and postulates.,, and intersect at point. xplain 2 Using the Inscribed ngle Theorem In the xplore you looked at the relationship between central angles and inscribed angles. Those results, combined with the definitions of arc measure, lead to the following theorem about inscribed angles and their intercepted arcs. n intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. Inscribed ngle Theorem The measure of an inscribed angle is equal to half the measure of its intercepted arc. m = 1_ 2 m xample 2 Use the Inscribed ngle Theorem to find inscribed angle measures. Houghton Mifflin Harcourt Publishing ompany etermine m, m, m, and m using the appropriate theorems and postulates. y the Inscribed ngle Theorem, m = 1 2 m, and so m = 2 54 = 108. y the rc ddition Postulate, m = m + m = = 126. y the Inscribed ngle Theorem, m = 1 2 m = = 63. Note that is a 2 semicircle, and so m = 180. y the Inscribed ngle Theorem, m = 1 2 m = = Module Lesson 1

6 etermine m WX, m XZ, m XWZ, and m WXZ using the appropriate theorems and postulates. y the Inscribed ngle Theorem, m WZX = m WX, and so m WX = 2 9 =. Note that WXZ is a therefore, m WXZ = 180. y the, and, W X Y Z m WXZ = m WX + m XZ and then m XZ = = y the, m XWZ = 1 2 m XZ = = Note that is a semicircle, and so m WYZ =. y the Inscribed ngle Theorem, m WXZ = 1 2 m = 1 2 =. Reflect 5. Iscussion xplain an alternative method for determining m XZ in xample Justify Reasoning How does the measure of compare to the measure of? xplain your reasoning. Your Turn 7. If m = 15, determine m using the appropriate theorems and postulates. 44 Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

7 xplain 3 Investigating Inscribed ngles on iameters You can examine angles that are inscribed in a semicircle. xample 3 onstruct and analyze an angle inscribed in a semicircle. Use a compass to draw a circle with center. Use a straightedge to draw a diameter of the circle. Label the diameter _. Use a straightedge to draw an inscribed angle on your circle from Step whose sides contain the endpoints of the diameter. Use a protractor to determine the measure of (to the nearest degree). Record the results in the table. ngle Measure ircle ircle 2 ircle 3 ircle 4 m Repeat the process three more times. Make sure to vary the size of the circle, and the location of the vertex of the inscribed angle. Record the results in the table in Part. xamine the results, and make a conjecture about the measure of an angle inscribed in a semicircle. Houghton Mifflin Harcourt Publishing ompany How can does the Inscribed ngle Theorem justify your conjecture? Module Lesson 1

8 Inscribed ngle of a iameter Theorem The endpoints of a diameter lie on an inscribed angle if and only if the inscribed angle is a right angle. Reflect 8. right angle is inscribed in a circle. If the endpoints of its intercepted arc are connected by a segment, must the segment pass through the center of the circle? laborate 9. n equilateral triangle is inscribed in a circle. How does the relationship between the measures of the inscribed angles and intercepted arcs help determine the measure of each angle of the triangle? 10. ssential Question heck-in What is the relationship between inscribed angles and central angles in a circle? Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

9 valuate: Homework and Practice Identify the chord (s), inscribed angle (s), and central angle (s) in the figure. The center of the circles in xercises 1, 2, and 4 is. Online Homework Hints and Help xtra Practice S T R U hord(s): Inscribedngle(s): entral ngle(s): hord(s): Inscribed ngle(s): entral ngle(s): 3. hord(s): Inscribed ngle(s): entral ngle(s): G 4. hord(s): Inscribed ngle(s): entral ngle(s): Houghton Mifflin Harcourt Publishing ompany In circle, m = 84. ind each measure. 5. m G 6. m H 84 G Module Lesson 1

10 The center of the circle is. ind each measure using the appropriate theorems and postulates. 7. m m 9. m 10. m ind each measure using the appropriate theorems and postulates. m = m 12. m Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

11 The center of the circle is. ind each measure using the appropriate theorems and postulates. m LM = 70 and m NP = L M P 60 N 13. m MNP 14. m LMN The center of the circle is O. ind each arc or angle measure using the appropriate theorems and postulates O 15. m 16. m Houghton Mifflin Harcourt Publishing ompany 17. m 18. m Module Lesson 1

12 Represent Real-World Problems Use the following information for xercises The circle graph shows how a typical household spends money on energy. Use the graph to find the measure of each arc. Heating and cooling 45% Q R Water heater 11% Home nergy Use P Others 19% V Lighting U 7% Washer and dryer 10% T ishwasher S 2% Refrigerator 6% 19. m PQ 20. m UPT 21. ommunicate Mathematical Ideas carpenter s square is a tool that is used to draw right angles. Suppose you are building a toy car and you have four small circles of wood that will serve as the wheels. You need to drill a hole in the center of each wheel for the axle. xplain how you can use the carpenter s square to find the center of each wheel arpenter s square hoose the expressions that are equivalent to m O. Select all that apply.. 1_ 2 m. m O. m. m. 2m. m G. 2m H. m O Houghton Mifflin Harcourt Publishing ompany Zoran Zeremski/ Shutterstock Module Lesson 1

13 23. In circle, the measures of,, and are in the ratio 3:4:5. ind m. 24. nalyze Relationships raw arrows to connect the concepts shown in the boxes. Then explain how the terms shown in the concept map are related. entral ngle hord rc ircle Inscribed ngle Inscribed ngle of a iameter Houghton Mifflin Harcourt Publishing ompany H.O.T. ocus on Higher Order Thinking 25. xplain the rror The center of the circle is G. elow is a student s work to find the value of x. xplain the error and find the correct value of x. _ is a diameter, so m = 180. Since m = m + m + m, m + m + m = x x = x = 90 5x (16x - 5) G 15x x = 4.5 Module Lesson 1

14 26. Multi-Step n inscribed angle with a diameter as a side has measure x. If the ratio of m to m is 1:4, what is m? x 27. Justify Reasoning To prove the Inscribed ngle Theorem you need to prove three cases. In ase 1, the center of the circle is on a side of the inscribed angle. In ase 2, the center the circle is in the interior of the inscribed angle. In ase 3, the center the circle is in the exterior of the inscribed angle. a. ill in the blanks in the proof for ase 1 to show that m = 1_ 2 m. Given: is inscribed in circle. Prove: m = 1_ 2 m Proof: Let m = x. raw _. is. So m = m by the Isosceles Triangle Theorem. Then = 2x by the xterior ngle Theorem. So, m = by the definition of the measure of an arc of a circle. Since m = and m =, m = 1_ 2. Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

15 b. raw and label a diagram for ase 2. Then use a paragraph proof to prove that the inscribed angle is one-half the intercepted arc. c. raw and label a diagram for ase 3. Then use a paragraph proof to prove that the inscribed angle is one-half the intercepted arc. Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

16 Lesson Performance Task iana arrives late at the theater for a play. Her ticket entitles her to sit anywhere in ircle G. She had hoped to sit in Seat, which she thought would give her the widest viewing angle of the stage. ut Seat is taken, as are all the other nearby seats in ircle G. The seating chart for the theater is shown. ircle K ircle G ircle Stage Identify two other spots where iana can sit that will give her the same viewing angle she would have had in Seat. xplain how you know how your points would provide the same viewing angle, and support your claim by showing the viewing angles on the drawing. Houghton Mifflin Harcourt Publishing ompany Module Lesson 1