Work with a partner. Use dynamic geometry software.
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1 10.4 Inscribed ngles and Polygons ssential uestion How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? n inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. n arc that lies between two lines, rays, or segments is called an intercepted arc. ecall that a polygon is an inscribed polygon when all its vertices lie on a circle. central angle intercepted arc P inscribed angle O Inscribed ngles and entral ngles Work with a partner. Use dynamic geometry software. NIN O PIION o be proficient in math, you need to communicate precisely with others. a. onstruct an inscribed angle in a circle. hen construct the corresponding central angle. b. Measure both angles. How is the inscribed angle related to its intercepted arc? c. epeat parts (a) and (b) several times. ecord your results in a table. Write a conjecture about how an inscribed angle is related to its intercepted arc. ample uadrilateral with Inscribed ngles Work with a partner. Use dynamic geometry software. a. onstruct a quadrilateral with each vertex on a circle. ample b. Measure all four angles. What relationships do you notice? c. epeat parts (a) and (b) several times. ecord your results in a table. hen write a conjecture that summarizes the data. ommunicate Your nswer 3. How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? 4. uadrilateral H is inscribed in, and m = 80. What is m? xplain. ection 10.4 Inscribed ngles and Polygons 597
2 10.4 Lesson What You Will Learn ore Vocabulary inscribed angle, p. 598 intercepted arc, p. 598 subtend, p. 598 Previous inscribed polygon circumscribed circle Use inscribed angles. Use inscribed polygons. Using Inscribed ngles ore oncept Inscribed ngle and Intercepted rc n inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. n arc that lies between two lines, rays, or segments is called an intercepted arc. If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to subtend the angle. inscribed angle intercepts. subtends. subtends. intercepted arc heorem Measure of an Inscribed ngle heorem he measure of an inscribed angle is one-half the measure of its intercepted arc. Proof x. 35, p. 604 m = 1 2 m he proof of the Measure of an Inscribed ngle heorem involves three cases. ase 1 enter is on a side of the inscribed angle. ase 2 enter is inside the inscribed angle. ase 3 enter is outside the inscribed angle. ind the indicated measure. a. m b. m Using Inscribed ngles 48 OLUION a. m = 1 2 m = 1 (48 ) = 24 2 b. m = 2m = 2 50 = 100 ecause is a semicircle, m = 180 m = = 80. P hapter 10 ircles
3 inding the Measure of an Intercepted rc ind m and m. What do you notice about and U? OLUION rom the Measure of an Inscribed ngle heorem, you know that m = 2m U = 2(31 ) = 62. lso, m = 1 2 m = 1 (62 ) = U o, U. xample 2 suggests the Inscribed ngles of a ircle heorem. heorem Inscribed ngles of a ircle heorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Proof x. 36, p. 604 inding the Measure of an ngle iven m = 75, find m. 75 OLUION oth and intercept H. o, by the Inscribed ngles of a ircle heorem. o, m = m = 75. Monitoring Progress ind the measure of the red arc or angle. H Help in nglish and panish at igideasmath.com 1. H V 38 U 3. Y 72 X W Z ection 10.4 Inscribed ngles and Polygons 599
4 Using Inscribed Polygons ecall that a polygon is an inscribed polygon when all its vertices lie on a circle. he circle that contains the vertices is a circumscribed circle. heorems Inscribed ight riangle heorem If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Proof x. 37, p. 604 inscribed polygon circumscribed circle m = 90 if and only if is a diameter of the circle. Inscribed uadrilateral heorem quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Proof x. 38, p. 604; igideasmath.com,,, and lie on if and only if m + m = m + m = 180. Using Inscribed Polygons ind the value of each variable. a. 2x b. z y OLUION a. is a diameter. o, is a right angle, and m = 90 by the Inscribed ight riangle heorem. 2x = 90 x = 45 he value of x is 45. b. is inscribed in a circle, so opposite angles are supplementary by the Inscribed uadrilateral heorem. m + m = 180 m + m = 180 z + 80 = y = 180 z = 100 y = 60 he value of z is 100 and the value of y is hapter 10 ircles
5 Using a ircumscribed ircle Your camera has a 90 field of vision, and you want to photograph the front of a statue. You stand at a location in which the front of the statue is all that appears in your camera s field of vision, as shown. You want to change your location. Where else can you stand so that the front of the statue is all that appears in your camera s field of vision? OLUION rom the Inscribed ight riangle heorem, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. o, draw the circle that has the front of the statue as a diameter. he statue fits perfectly within your camera s 90 field of vision from any point on the semicircle in front of the statue. Monitoring Progress Help in nglish and panish at igideasmath.com ind the value of each variable. 4. L x y K 40 M 5. 2x y x c 10x (2c 6) y 82 8x V U 8. In xample 5, explain how to find locations where the left side of the statue is all that appears in your camera s field of vision. ection 10.4 Inscribed ngles and Polygons 601
6 10.4 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. VOULY n arc that lies between two lines, rays, or segments is called a(n). 2. IN WO, M UION Which is different? ind both answers. ind m. ind m ind m. ind m. Monitoring Progress and Modeling with Mathematics In xercises 3 8, find the indicated measure. (ee xamples 1 and 2.) 3. m 4. m m N 6. m N 7. m VU 30 V 160 L M U 8. m WX W X Y In xercises 9 and 10, name two pairs of congruent angles W X Y Z In xercises 11 and 12, find the measure of the red arc or angle. (ee xample 3.) 11. H P In xercises 13 16, find the value of each variable. (ee xample 4.) x y 95 J 54 M 4b K 3a L m 2k 16. X Y 3x 2y 17. O NLYI escribe and correct the error in finding m. 53 m = 53 Z 602 hapter 10 ircles
7 18. MOLIN WIH MHMI carpenter s square is an L-shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use the carpenter s square to draw a diameter on the circular piece of wood? (ee xample 5.) ONIN In xercises 23 28, determine whether a quadrilateral of the given type can always be inscribed inside a circle. xplain your reasoning. 23. square 24. rectangle 25. parallelogram 26. kite 27. rhombus 28. isosceles trapezoid MHMIL ONNION In xercises 19 21, find the values of x and y. hen find the measures of the interior angles of the polygon x 26y 21y 2x 29. MOLIN WIH MHMI hree moons,,, and, are in the same circular orbit 100,000 kilometers above the surface of a planet. he planet is 20,000 kilometers in diameter and m = 90. raw a diagram of the situation. How far is moon from moon? 30. MOLIN WIH MHMI t the movie theater, you want to choose a seat that has the best viewing angle, so that you can be close to the screen and still see the whole screen without moving your eyes. You previously decided that seat 7 has the best viewing angle, but this time someone else is already sitting there. Where else can you sit so that your seat has the same viewing angle as seat 7? xplain. movie screen 20. 9y 24y 14x 4x ow ow ow ow ow ow ow y 6y 31. WIIN right triangle is inscribed in a circle, and the radius of the circle is given. xplain how to find the length of the hypotenuse. 2x 4x 32. HOW O YOU I? Let point Y represent your location on the soccer field below. What type of angle is Y if you stand anywhere on the circle except at point or point? 22. MKIN N UMN Your friend claims that P P P. Is your friend correct? xplain your reasoning. P ection 10.4 Inscribed ngles and Polygons 603
8 33. WIIN xplain why the diagonals of a rectangle inscribed in a circle are diameters of the circle. 34. HOUH POVOKIN he figure shows a circle that is circumscribed about. Is it possible to circumscribe a circle about any triangle? Justify your answer. 35. POVIN HOM If an angle is inscribed in, the center can be on a side of the inscribed angle, inside the inscribed angle, or outside the inscribed angle. Prove each case of the Measure of an Inscribed ngle heorem. a. ase 1 iven is inscribed in. Let m = x. enter lies on. Prove m = 1 2 m (Hint: how that is isosceles. hen write m in terms of x.) b. ase 2 Use the diagram and auxiliary line to write iven and Prove statements for ase 2. hen write a proof. c. ase 3 Use the diagram and auxiliary line to write iven and Prove statements for ase 3. hen write a proof. 36. POVIN HOM Write a paragraph proof of the Inscribed ngles of a ircle heorem. irst, draw a diagram and write iven and Prove statements. x 37. POVIN HOM he Inscribed ight riangle heorem is written as a conditional statement and its converse. Write a plan for proof for each statement. 38. POVIN HOM opy and complete the paragraph proof for one part of the Inscribed uadrilateral heorem. iven with inscribed quadrilateral Prove m + m = 180, m + m = 180 y the rc ddition Postulate, m + = 360 and m + m = 360. Using the heorem, m = 2m, m = 2m, m = 2m, and m = 2m. y the ubstitution Property of quality, 2m + = 360, so. imilarly,. 39. IIL HINKIN In the diagram, is a right angle. If you draw the smallest possible circle through tangent to, the circle will intersect at J and at K. ind the exact length of JK IIL HINKIN You are making a circular cutting board. L o begin, you glue eight 1-inch boards together, as shown. J hen you draw and cut a circle with an 8-inch diameter from M the boards. K a. H is a diameter of the circular cutting board. Write a proportion relating J and JH. tate a theorem to justify your answer. b. ind J, JH, and J. What is the length of the cutting board seam labeled K? 5 4 H Maintaining Mathematical Proficiency olve the equation. heck your solution. (kills eview Handbook) x = x = = 2x = 1 (x 30) 2 eviewing what you learned in previous grades and lessons 604 hapter 10 ircles
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