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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Name Class Date 15.1 Central Angles and Inscribed Angles Essential Question: How can you determine the measures of central angles and inscribed angles of a circle? Explore Resource Locker Investigating Central Angles and Inscribed Angles A chord is a segment whose endpoints lie on a circle. A central angle is an angle less than 180° whose vertex lies at the center of a circle. An 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. Chords Central Angle A A _ _ AB and AB B C B ∠ACD Inscribed Angle C D ∠ACD D © Houghton Mifflin Harcourt Publishing Company AUse a compass to draw a circle. Label the center C. B D E Use a straightedge to draw an acute inscribed angle on your circle from Step A. Label the angle as ∠DEF. C Check students’ drawings. CUse a straightedge to draw the corresponding central angle, ∠DCF. F Check students’ drawings. D 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. Answers may vary. Angle Measure Circle C Circle 2 Circle 3 Circle 4 Circle 5 Circle 6 Circle 7 m∠DEF 32° 51° 75° 90° 127° 155° 173° m∠DCF 64° 102° 150° 180° 106° 50° 14° 360° - m∠DCF 296° 258° 210° 180° 254° 310° 346° Module 15 GE_MNLESE385801_U6M15L1.indd 779 779 Lesson 1 02/04/14 11:30 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A E DO NOT Correcti Repeat Steps A-D six more times. Examine a variety of inscribed angles (two more acute, one right, and three obtuse). Record your results in the table in Step D. Answers may vary. Possible answers are given. Reflect 1. Examine 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. For inscribed angles with measures less than or equal to 90°, the measure of the inscribed angle is equal to half the measure of the corresponding central angle. 2. Examine 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. For inscribed angles with measures greater than or equal to 90°, the measure of the inscribed angle is equal to half the difference between 360° and the measure of the corresponding central angle. Explain 1 Understanding Arcs and Arc Measure An 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. Arc A minor arc is an arc whose points are on or in the interior of a corresponding central angle. Measure Figure The measure of a minor arc is equal to the measure of the central angle. A ⁀ = m∠ACB mAB ⁀ AB C B The measure of a major arc is equal to 360° minus the measure of the central angle. ⁀ = 360°-m∠ACB mADB A semicircle is an arc whose endpoints are the endpoints of a diameter. The measure of a semicircle is 180°. © Houghton Mifflin Harcourt Publishing Company A major arc is an arc whose points are on or in the exterior of a corresponding central angle. A D ⁀ ADB C B A ⁀ ADB D C B Module 15 GE_MNLESE385801_U6M15L1.indd 780 780 Lesson 1 22/03/14 11:17 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Adjacent arcs are arcs of the same circle that intersect in exactly one ⁀ and EF ⁀ are adjacent arcs. point. DE E D F C Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. ⁀ =m AD ⁀ +m DB ⁀ mADB D A B C Example 1 ⁀ using the ⁀ = 33°, determine m ABD If m∠BCD = 18° and mEF ‹ › ‹ › − − appropriate theorems and postulates. AF and BE intersect at Point C. ⁀ = 33°, then m∠ECF = 33°. If m∠ECF = 33°, then Ifm EF m∠ACB = 33° by the Vertical Angles Theorem. If m∠ACB = 33° ⁀ = 33° and mBD ⁀ = 18°. By the Arc and m∠BCD = 18°, then mAB ⁀ =m AB ⁀ +m BD ⁀ = 51°. ⁀, and so mABD Addition Postulate, mABD A B E C F ⁀ = 27°, determine mNP ⁀ using the appropriate theorems and JK ‹ › ‹ › − − postulates. MK and NJ intersect at Point C. D © Houghton Mifflin Harcourt Publishing Company Ifm ⁀ = 27°, then m∠JCK = 27°. If m∠JCK = 27°, then JK Ifm m∠ MCN = 27° by the Vertical Angles Theorem . If m ⁀ = 27° ∠MCN = 27° and m∠MCP = 90 °, then mMN andm ⁀ = 90 °. By the Arc Addition Postulate , MNP m M N ⁀ =m MN ⁀ +m NP ⁀, and so mNP = m MNP Module15 GE_MNLESE385801_U6M15L1.indd 781 ⁀ MNP 781 P C J K ⁀ = 63 ° - mMN Lesson1 22/03/14 11:17 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti Reflect 3. The minute hand of a clock sweeps out an arc as time moves forward. From 3:10 p.m. to 3:30 p.m., what is the measure of this arc? Explain your reasoning. 11 12 1 2 10 6 2 3 9 8 4 7 1 10 3 9 8 12 11 5 4 7 3:10 6 5 3:30 1 A complete rotation corresponds to 60 minutes, so 20 minutes is __ of a complete rotation. 3 1 Since __ × 360° = 120°, the measure of the arc is 120°. 3 Your Turn 4. ⁀ = 45° and m∠ACD = 56°, determine mBD ⁀ using the appropriate If mEF ‹ › − ‹ › ‹ › − − theorems and postulates. AE , BF , and DC intersect at Point C. D E F B C A ⁀ = 45°, then m∠ECF = 45°. If m∠ECF = 45°, then m∠ACB = 45° by the Vertical If mEF ⁀= ⁀ = 45° and mABD Angles Theorem. If then m∠ACB = 45° and m∠ACD = 56°, then mAB Explain 2 Using the Inscribed Angle Theorem In the Explore 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 . An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the vpoints of the circle between them. Inscribed Angle Theorem The measure of an inscribed angle is equal to half the measure of its intercepted arc. 1 mAB ⁀ m∠ADB = _ 2 A D C © Houghton Mifflin Harcourt Publishing Company ⁀ = mAB ⁀ + mBD ⁀ , and so mBD ⁀ = 11°. 56°. By the Arc Addition Postulate, mABD B Module 15 GE_MNLESE385801_U6M15L1.indd 782 782 Lesson 1 22/03/14 11:17 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Example 2 Use the Inscribed Angle Theorem to find inscribed angle measures. ⁀,m BD ⁀, m∠DAB, and m∠ADE using the appropriate DE theorems and postulates. Determinem By the Inscribed Angle Theorem, B 18° E ⁀, and so m∠DAE = __12 mDE ⁀ = 2 × 54° = 108°. By the Arc Addition Postulate, mDE ⁀ =m BE ⁀ +m ED ⁀ = 18°+ 108° = 126°. By the Inscribed Angle mBD C 54° A ⁀ is a ⁀ = __1 × 126° = 63°. Note that ADE Theorem, m∠DAB = __12 mBD 2 D ⁀ = 180°. By the Inscribed Angle Theorem, semicircle, and so mADE m∠ADE = __12 mADE = __12 × 180° = 90°. ⁀, m∠XWZ, and m∠WXZ using the appropriate ⁀,m XZ WX theorems and postulates. Determinem By the Inscribed Angle Theorem, m and so m therefore,m X W ⁀ is a semicircle and, ⁀ = 2 × 9° = 18° . Note that WXZ WX C Z 9° Y 106° ⁀ = 180°. By the Arc Addition Postulate , WXZ ⁀ =m WX ⁀ and then mXZ ⁀ = 180° - 18° = 162° ⁀ +m XZ WXZ m By the, Note that Inscribed Angle Theorem m∠XWZ = __1 mXZ ⁀ = __1 × 162° = 81°. 2 2 ⁀ WYZ Angle Theorem, m © Houghton Mifflin Harcourt Publishing Company ∠WZX = _1 mWX, 2 ⁀ = 180° . By the Inscribed is a semicircle, and so mWYZ ∠WZX = __12 m ⁀ WYZ = __12 × 180° = 90° . Reflect 5. ⁀ in Example 2B. DIscussion Explain an alternative method for determining m∠ XZ Use the Inscribed Angle Theorem to determine m∠WXZ = 90°. The angle measures in a triangle sum to 180°, therefore m∠XWZ = 81° and by the Inscribed Angles Theorem, m⁀ XZ = 162°. 6. Justify Reasoning How does the measure of ∠ABD compare to the measure of ∠ACD? Explain your reasoning. AD . Since the measure m∠ABD = m∠ACD; Each angle intercepts ⁀ A D of each angle is equal to half the measure of ⁀ AD , the measures of ∠ABD and ∠ACD are the same. B C Module15 GE_MNLESE385801_U6M15L1.indd 783 783 Lesson1 22/03/14 11:17 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti Your Turn 7. If m∠EDF = 15°, determine m∠ABE using the appropriate theorems and postulates. 44° A Since ∠EDF and ∠EBF intercept the same arc, they have the same F C measure. By the Inscribed Angle Theorem, 1 ⁀ __ m∠ABF = __ mAF = 12 × 44° = 22°. By the Angle Addition 2 E B Postulate, D m∠ABE = m∠ABF + m∠EBF. Therefore, m∠ABE = 22° + 15° = 37°. Explain 3 Investigating Inscribed Angles on Diameters You can examine angles that are inscribed in a semicircle. Example 3 Construct and analyze an angle inscribed in a semicircle. A Use a compass to draw a circle with center C. Use a straightedge to draw a _ diameter of the circle. Label the diameter DF. B Use a straightedge to draw an inscribed angle ∠DEF on your circle from Step A whose sides contain the endpoints of the diameter. E D F C Check students’ drawings. C Use a protractor to determine the measure of ∠DEF (to the nearest degree). Record the results in the table. Circle C Circle 2 Circle 3 Circle 4 m∠DEF 90° 90° 90° 90° D 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 C. E Examine the results, and make a conjecture about the measure of an angle inscribed in a semicircle. © Houghton Mifflin Harcourt Publishing Company Angle Measure The measure of an angle inscribed in a semicircle is 90°. F How can does the Inscribed Angle Theorem justify your conjecture? The Inscribed Angle Theorem states that the measure of an inscribed angle is equal to half the measure of its intercepted arc. The arc formed by a diameter is a semicircle with a measure equal to 180°. Therefore, the measure of the inscribed angle is 90°. Module 15 GE_MNLESE385801_U6M15L1.indd 784 784 Lesson 1 22/03/14 11:17 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Inscribed Angle of a Diameter Theorem The endpoints of a diameter lie on an inscribed angle if and only if the inscribed angle is a right angle. Reflect 8. A 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? Yes, the endpoints of the intercepted arc lie on a diameter. And by definition, a diameter passes through the center of the circle. Elaborate 9. An 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? © Houghton Mifflin Harcourt Publishing Company Together the three angles intercept the entire circle, which has a measure of 360°. Since the angles of an equilateral triangle are congruent, they each intercept an arc of 120°. The measure of an inscribed angle is equal to half the measure of its intercepted arc, so each angle of the triangle measures 60°. 10. Essential Question Check-In What is the relationship between inscribed angles and central angles in a circle? If an inscribed angle is acute, its measure is equal to half the measure of its associated central angle. If an inscribed angle is obtuse, its measure is equal to half the difference between the measure of its associated central angle and 360°. Module 15 GE_MNLESE385801_U6M15L1.indd 785 785 Lesson 1 22/03/14 11:17 PM