MODULE. (40 + 8x) + (5x 16) = 180. STUDY GUIDE REVIEW Angles and Segments in Circles. Key Vocabulary


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1 STUDY GUIDE REVIEW Angles and Segments in ircles ODULE 15 Essential Question: How can you use angles and segments in circles to solve realworld problems? EY EXALE (Lesson 15.1) Determine m DE, m BD, m DAB, and m ADE. Since chord AE passes through the center of the circle at, the chord AE is a diameter of the circle. ABE is then a semicircle, and m ABE = 180. But m ADE = 1 2 m ABE = 90. ΔADE is a right triangle with m AED = 36. (40 +8x) J A 54 L B 18 Also, m DAE = 1 2 m DE, which implies that m DE = 2m DAE = 108. Since m BD = m BE + m DE, m BD = = 126. Finally, m DAB = 1 2 m BD = 63. EY EXALE (Lesson 15.2) Determine the angles J,, L, and in the given quadrilateral. (40 + 8x) + (5x 16) = x = x = 156. x = 12. m J = x = (12) = 136 m L = 5x  16 = 5 (12)  16 = = (12) m = _ = 60. m + m = m = = 120 D (5x 16) E 20x 4 ey Vocabulary chord (cuerda) central angle (ángulo central) inscribed angle (ángulo inscrito) arc (arco) minor arc (arco menor) major arc (arco principal) semicircle (semicirculo) adjacent arcs (arcos adyacentes) Inscribed Angle Theorem (teorema del ángulo inscrito) Inscribed Quadrilateral Theorem (teorema del ángulo inscrito) tangent (tangent) point of tangency (punto de tangencia) TangentRadius Theorem (teorema de la tangenteradio) hordhord roduct Theorem (teorema del producto de la cuerda de la cuerda) secant (secante) secant segment (segmente secante) external secant segment (segmento externo secante) SecantSecant Theorem (Teorema de la secantesecante) tangent segment (segmento tangente) Intersecting hords Angle easure Theorem (teorema de medida de ángulo de intersección acordes) TangentSecant Interior Angle easure Theorem (teorema de la medida de ángulo interior tangentesecante) TangentSecant Exterior Angle easure Theorem (teorema de la medida de ángulo exterior tangentesecante) odule Study Guide Review
2 EY EXALE (Lesson 15.3) Two tangent lines are drawn to a circle from point intersecting the circle at points and N. If m N = 210, what is m N? N If m N = 210, then m N = = 150. m N = 1_ 2 (m N  m N ) m N = 1_ ( ) = 1_ (60 ) = EY EXALE (Lesson 15.4) A tangent and a secant are drawn to a circle from the external point B. The point of tangency is at point A, and the secant intersects the circle at points and D. Find x. From the SecantTangent roduct Theorem, we can say that BD B = AB 2. So (2 + x) 2 = 8 2. A B 8 2 x D So 4 + 2x = 64. 2x = 60 and x = 30. EY EXALE (Lesson 15.5) Two chords intersect the interior of a circle at point T. Find m R. By the Intersecting hords Angle easure Theorem we can say the following: m QTS = 1_ 2 (m QS + m R ) 60 = 1_ 2 (80 + m R ) 120 = (80 + m R ) = m R 40 = m R R T 60 Q 80 S odule Study Guide Review
3 EXERISES Use the Inscribed Angle Theorem. (Lesson 15.1) 1. Find the measure of the intercepted arc for an inscribed angle of 50. Use the Inscribed Quadrilateral Theorem. (Lesson 15.2) 2. If one angle of a quadrilateral inscribed in a circle is 50, what is the measure of its opposite angle? Use the ircumscribed Angle Theorem. (Lesson 15.3) 3. Two tangents are drawn from an external point A to a circle. If one of the intercepted arcs on the circle is 120, what must be the measure of the other intercepted arc? Use the hordhord roduct Theorem. (Lesson 15.4) R T S Q Given RT = 2, TS = 6, and T = 3. Find TQ Use the TangentSecant Exterior Angle easure Theorem. (Lesson 15.5) L Find m L in the diagram given the m and m JN. J N 83 odule Study Guide Review
4 ODULE ERFORANE TAS How any arbles Will Fit? onsider a package of marbles in the shape of a triangular prism. The crosssection of the package is an equilateral triangle with a side length of 1.5 inches, and the length of the package is 10 inches. What is the diameter of the largest marble that will fit inside the package? How many such marbles can fit within the package? Start by listing in the space below how you plan to tackle the problem. Then use your own paper to complete the task. Be sure to write down all your data and assumptions. Then use words, numbers, diagrams, or algebra to explain how you reached your conclusion. odule Study Guide Review
5 Ready to Go On? Angles and Segments in ircles 1. If m BD = 20 and m EF = 34, determine m ABD using the appropriate theorems and postulates. (Lesson 15.1) If m EF = 34, then m EF =. If m EF = 34, then m AB = by the Theorem. If m AB = 34 and m BD = 20, then Online Homework Hints and Help Extra ractice A m AB = and. By the so m ABD =., m ABD = m AB + m BD, and 2. Find the measures of each angle in the inscribed quadrilateral. (Lesson 15.2) E F D B (y ) Q (5y + 4) R (15y + 16) S Fill in the proper conclusions based on known theorems and relationships. (Lesson 15.5) 3. Using the given figure where _ and _ N are tangent to the circle at and N respectively, what can you say about the following? a. What angles are right angles? b. Suppose that m N = 80. What is m N? ESSENTIAL QUESTION 4. What are the major theorems that allow you to determine the relationships between angles formed by lines that intersect a circle? N odule Study Guide Review
6 ODULE 15 IXED REVIEW Assessment Readiness SELETED RESONSE 1. An angle of 20 is inscribed in a circle. ould the given value be the measure of the arc intercepted by this angle? Select Yes or No for A. A. 10 Yes No B. 20 Yes No. 40 Yes No 2. The points A, B,, and D are taken in order on the circumference of a circle. hords A and BD intersect at point E. m AB = 76 and m D = 80. hoose True or False for each statement. A. m AB = 156 True False B. m AEB = 78 True False. m AED = 72 True False 3. Line BF bisects AB, m ABF = 6x, and m FB = (2x + 60). hoose True or False for each statement. A. m FB = 45 True False B. m AB = 180 True False. ABF is a right angle. True False 4. If two chords intersect inside a circle, then what do you know about the products of the lengths of the segments of the chords? How can you determine whether two circles are similar? 5. AB is inscribed in a circle such that vertices A and B lie on a diameter of the circle. If the length of the diameter of the circle is 13 and the length of chord B is 5, find side A. odule Study Guide Review
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