10.4 Inscribed Angles

1. 10.4 Inscribed Angles

2. Objectives • Find measures of inscribed angles • Find measures of angles of inscribed polygons

3. A C B Inscribed Angles • An inscribed angle is an angle that has its vertices on the circle and its sides are chords of the circle.

4. A C B Inscribed Angles • Theorem 10.5 (Inscribed Angle Theorem):The measure of an inscribed angle equals ½ the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). mACB = ½m or 2 mACB =

5. In and Find the measures of the numbered angles. Example 1:

6. First determine Example 1: Arc Addition Theorem Simplify. Subtract 168 from each side. Divide each side by 2.

7. So, m Example 1:

8. Answer: Example 1:

9. In and Find the measures of the numbered angles. Answer: Your Turn:

10. Inscribed Angles • Theorem 10.6:If two inscribed s intercept  arcs or the same arc, then the s are . mDAC  mCBD

11. PROBABILITYPoints M and N are on a circle so that . Suppose point L is randomly located on the same circle so that it does not coincide with M or N. What is the probability that Since the angle measure is twice the arc measure, inscribed must intercept , so L must lie on minor arc MN. Draw a figure and label any information you know. Example 3:

12. The probability that is the same as the probability of L being contained in . Answer: The probability that L is located on is Example 3:

13. PROBABILITYPoints A and X are on a circle so that Suppose point B is randomly located on the same circle so that it does not coincide with A or X. What is the probability that Answer: Your Turn:

14. o Angles of Inscribed Polygons • Theorem 10.7:If an inscribed  intercepts a semicircle, then the  is a right . i.e. If AC is a diameter of , then the mABC = 90°.

15. A B O D C Angles of Inscribed Polygons • Theorem 10.7:If a quadrilateral is inscribed in a , then its opposite s are supplementary. i.e. Quadrilateral ABCD is inscribed in O, thus A and C are supplementary and B and D are supplementary.

16. ALGEBRATriangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and Example 4:

17. are right triangles. since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so . Example 4: Angle Sum Theorem Simplify. Subtract 105 from each side. Divide each side by 3.

18. Use the value of x to find the measures of Answer: Example 4: Given Given

19. ALGEBRATriangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and Answer: Your Turn:

20. Quadrilateral QRST is inscribed in If and find and Draw a sketch of this situation. Example 5:

21. To find we need to know To find first find Example 5: Inscribed Angle Theorem Sum of angles in circle=360 Subtract 174 from each side.

22. To find we need to know but first we must find Example 5: Inscribed Angle Theorem Substitution Divide each side by 2. Inscribed Angle Theorem

23. Answer: Example 5: Sum of angles in circle=360 Subtract 204 from each side. Inscribed Angle Theorem Divide each side by 2.

24. Quadrilateral BCDE is inscribed in If and find and Answer: Your Turn: