1. Inscribed Angles

2. 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.

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

4. Example 1: a) Find m  X.

5. Example 1: c) Find m  C.

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

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

9. Example 3: Write a two-column proof. Given: Prove: ∆MNP  ∆LOP StatementsReasons 1. 2. 3. 4. Given If minor arcs are congruent, then corresponding chords are congruent. Definition of intercepted arc Inscribed angles of the same arc are congruent.  M   L

10. Example 3: Write a two-column proof. Given: Prove: ∆MNP  ∆LOP StatementsReasons 5. 6. Vertical angles are congruent. AAS Congruence Theorem  MPN   OPL ∆MNP  ∆LOP

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

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

15. Example 5: 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(14x) + (8x + 4)= 180 m  S = 9022x + 4= 180 22x= 176 x= 8 Answer: So, m  S = 90º and m  T = 8(8) + 4 or 68º.

16. Example 5: 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 (11x) + (3x + 12)= 180 14x + 12= 180 14x= 168 x= 12 Answer: So, m  N = 3(12) + 12 or 48º.