1. 5-Minute Check on Lesson 10-5 Transparency 10-6 Click the mouse button or press the Space Bar to display the answers. Determine whether each segment is tangent to the given circle. 1. BC 2. QR Find x. Assume that the segments that appear to be tangents are tangents. 3. 4. 5. What is the measure of PS? Standardized Test Practice: ACBD 101214 18 Yes No x = 12 x = 20√5 or 44.72 B

2. Lesson 10-6 Secants, Tangents, and Angle Measures

3. Objectives Find measures of angles formed by lines intersecting on or inside a circle Find measures of angles formed by lines intersecting outside a circle

4. Vocabulary Secant – a line that intersects a circle in exactly two points

5. y x Interior Angles formed by a Secant & a Tangent m  3 = ½ (m Arc JMK) m  4 = ½ (m Arc JLK) Circles – Secants Interior Angles formed by 2 Secants m  1 = ½ (m Arc MJ + m Arc LK) m  2 = ½ (m Arc MK + m Arc JL) Center J K L M 1 2 P 3 4

6. Circles – External Angles J J K K L T M M N J S T M m  J = ½(m Arc MN – m Arc LK) m  J = ½|m Arc MN – m Arc LK| m  J = ½(m Arc TM – m Arc TK) m  J = ½|m Arc TM – m Arc TK| m  J = ½(m Arc TMS – m Arc TS) m  J = ½|m Arc TMS – m Arc TS| Two SecantsTwo TangentsSecant & Tangent

7. Example 6-1a Find if and Method 1

8. Example 6-1b Method 2 Answer: 98

9. Example 6-1d Answer: 138 Find if and

10. Example 6-2a Find if and Answer: 55

11. Example 6-2c Answer: 58 Find if and

12. Example 6-3a Find x. Theorem 10.14 Multiply each side by 2. Add x to each side. Subtract 124 from each side. Answer: 17

13. Example 6-3c Find x. Answer: 111

14. Example 6-4a JEWELRY A jeweler wants to craft a pendant with the shape shown. Use the figure to determine the measure of the arc at the bottom of the pendant. Let x represent the measure of the arc at the bottom of the pendant. Then the arc at the top of the circle will be 360 – x. The measure of the angle marked 40° is equal to 1/2 the difference of the measure of the two intercepted arcs. Multiply each side by 2 and simplify. Add 360 to each side. Divide each side by 2. Answer: 220

15. Example 6-4c Answer: 75 PARKS Two sides of a fence to be built around a circular garden in a park are shown. Use the figure to determine the measure of

16. Example 6-5a Find x. Multiply each side by 2. Add 40 to each side. Divide each side by 6. Answer: 25

17. Example 6-5c Find x. Answer: 9

18. Summary & Homework Summary: –The measure of an angle formed by two secant lines is half the positive difference of its intercepted arcs –The measure of angle formed by a secant and tangent line is half its intercepted arc Homework: –pg 564-566; 12-14, 18-20; 23-24, 26, 29, 34-36