11.5 Inscribed Angles and
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1 age 1 of Inscribed ngles and olygons Goal Use properties of inscribed angles. ey Words inscribed angle intercepted arc inscribed circumscribed n inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. ctivity 11.5 shows the relationship between an inscribed angle and its intercepted arc. inscribed angle intercepted arc THO 11.7 easure of an Inscribed ngle Words If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. Symbols ma 1 2 ms X 1 Find easures of Inscribed ngles and rcs Find the measure of the inscribed angle or the intercepted arc. a. N b. W IStudent Help I S S Z O N. O O XS ore examples at classzone.com Solution a. man 1 2 mns 1 2 () 50 b. mazyx 1 2 mzwx t The measure of an inscribed angle is half the measure of its intercepted arc. Substitute for mns. Simplify. Z 105 The measure of an inscribed angle is half the measure of its intercepted arc. Y X mzwx t Substitute 105 for mazyx. 210 mzwx t ultiply each side by hapter 11 ircles
2 age 2 of 6 Find easures of Inscribed ngles and rcs Find the measure of the inscribed angle or the intercepted arc F N 120 Inscribed and ircumscribed If all the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle. inscribed triangle circumscribed circle inscribed quadrilateral THO 11.8 Words If a triangle inscribed in a circle is a right triangle, then the hypotenuse is a diameter of the circle. If a side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle. X 2 Find ngle easures Student Help OO To review the orollary of the Triangle Sum Theorem, see p Find the values of x and y. Solution ecause T is inscribed in a circle and &* is a diameter, it follows from Theorem 11.8 that T is a right triangle with hypotenuse &*. Therefore, x 90. ecause a and a are acute angles of a right triangle, y Inscribed ngles and olygons 615
3 age 3 of 6 Find ngle easures Find the values of x and y in Visualize It! a and af are opposite angles. a and ag are opposite angles. F THO 11.9 Words If a quadrilateral can be inscribed in a circle, then its opposite angles are supplementary. If the opposite angles of a quadrilateral are supplementary, then the quadrilateral can be inscribed in a circle. F G G X 3 Find ngle easures Find the values of y and z. Solution ecause STU is inscribed in a circle, by Theorem 11.9 opposite angles must be supplementary. U z y S 120 T as and au are opposite angles. a and at are opposite angles. mas mau 1 ma mat z 1 y 60 z 100 Find ngle easures Find the values of x and y in hapter 11 ircles
4 age 4 of xercises Guided ractice Vocabulary heck In xercises 1 and 2, use the diagram at the right. 1. Name the inscribed angles. 2. Identify the two pairs of opposite angles in the inscribed quadrilateral. Skill heck Find the measure of the blue intercepted arc Find the value of each variable y z 85 ractice and pplications xtra ractice See p ngle easures Find the measure of the inscribed angle Homework Help xample 1: xs xample 2: xs xample 3: xs N U T 238 S 11.5 Inscribed ngles and olygons 617
5 age 5 of 6 rc easures Find the measure of the blue intercepted arc S T U N X W 103 Y Z Student Help VISU STTGY In xs , copy the diagram and add information to it as you solve the exercises, as shown on p rc and ngle easures In xercises 21 26, use the diagram below to find the intercepted arc or inscribed angle. 21. mr 22. ma 23. ma 24. mr 25. ma 26. mr 27. re T and T similar? xplain your reasoning. 47 Inscribed ight Triangles Find the value of each variable. xplain your reasoning S arpenter s Square carpenter s square is an -shaped tool used to draw right angles. Suppose you are making a toy truck. To make the wheels you trace a circle on a piece of wood. How could you use a carpenter s square to find the center of the circle? IStudent Help I S S Z O N. O HOWO H xtra help with problem solving in x. 31 is at classzone.com 618 hapter 11 ircles
6 age 6 of 6 Inscribed Quadrilaterals Find the values of x and y You be the udge an the quadrilateral always be inscribed in a circle? xplain your answer. 35. square 36. isosceles trapezoid 37. rhombus 38. rectangle Standardized Test ractice 39. ultiple hoice In the diagram at the right, if a is a central angle and ma, what is ma? ultiple hoice In the diagram at the right, what are the values of x and y? F x 80, y 95 G x 85, y 100 H x 95, y 80 x 95, y ixed eview ultiplying adicals ultiply the radicals. Then simplify if possible. (esson 10.1) p p p (8 2 ) (3 3 ) p 1 0 Solving ight Triangles Solve the right triangle. ound decimals to the nearest tenth. (esson 10.6) lgebra Skills valuating xpressions valuate the expression when x 2. (Skills eview, p. 670) 50. 3x x x (x 4)(x 4) 54. x 2 3x x 3 x Inscribed ngles and olygons 619
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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
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