Inscribed Angles An inscribed angle has a vertex

Inscribed Angles

An inscribed angle has a vertex on a circle and sides that contain chords of the circle. In , �C, QRS is an inscribed angle. An intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle. In �C, minor arc is intercepted by QRS.

There are three ways that an angle can be inscribed in a circle. For each of these cases, the following theorem holds true.

Example 1: a) Find m X.

Example 1: c) Find m C.

Example 2: a) Find m R. R m R 12 x – 13 x S = m S = 9 x + 2 =5 R and S both intercept. Definition of congruent angles Substitution Simplify. Answer: So, m R = 12(5) – 13 or 47º.

Example 2: b) Find m I. I m I 8 x + 9 x J = m J = 10 x – 1 =5 R and S both intercept. Definition of congruent angles Substitution Simplify. Answer: So, m I = 8(5) + 9 or 49º.

Example 3: a) Find m B. ΔABC is a right triangle because C inscribes a semicircle. m A + m B + m C = 180 (x + 4) + (8 x – 4) + 90 = 180 9 x = 90 x = 10 Angle Sum Theorem Substitution Simplify. Subtract 90 from each side. Divide each side by 9. Answer: So, m B = 8(10) – 4 or 76º.

Example 3: b) Find m D. ΔDEF is a right triangle because F inscribes a semicircle. m D + m E + m F = 180 (2 x + 6) + (8 x + 4) + 90 = 180 10 x + 100 = 180 10 x = 80 x =8 Angle Sum Theorem Substitution Simplify. Subtract 90 from each side. Divide each side by 10. Answer: So, m D = 2(8) + 6 or 22º.

Example 4: a) An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find m S and m T. Since TSUV is inscribed in a circle, opposite angles are supplementary. m S + m V = 180 m U + m T = 180 m S + 90 = 180 (14 x) + (8 x + 4) = 180 m S = 90 22 x + 4 = 180 22 x = 176 x =8 Answer: So, m S = 90º and m T = 8(8) + 4 or 68º.

Example 4: b) An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find m N. Since LMNO is inscribed in a circle, opposite angles are supplementary. m L + m N = 180 (11 x) + (3 x + 12) = 180 14 x + 12 = 180 14 x = 168 x = 12 Answer: So, m N = 3(12) + 12 or 48º.

Example 5: A square is inscribed in a circle. What is the ratio of the area of the circle to the area of the square? The triangle formed inside of the square uses radii of the circle and is a right isosceles triangle, therefore the side length of the square is. Thus, the area of the square is 2 r 2. The area of the circle is πr 2. r