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5-Minute Check on Lesson 10-5 Transparency 10-6 Determine whether each segment is tangent to the given circle. 1. BC 2. QR Yes No Find x. Assume that the segments that appear to be tangents are tangents. 3. 4. x = 20√5 or 44.72 x = 12 5. Standardized Test Practice: A 10 B 12 C What is the measure of PS? 14 D 18 Click the mouse button or press the Space Bar to display the answers. Lesson 10-6 Secants, Tangents, and Angle Measures 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 Vocabulary • Secant – a line that intersects a circle in exactly two points Circles – Secants y Interior Angles formed by a Secant & a Tangent m3 = ½ (m Arc JMK) m4 = ½ (m Arc JLK) M 3 1 2 4 K x J P Center Interior Angles formed by 2 Secants m1 = ½ (m Arc MJ + m Arc LK) m2 = ½ (m Arc MK + m Arc JL) L Circles – External Angles Two Secants Secant & Tangent J J L K J K T N Two Tangents M M M S T mJ = ½(m Arc TM – m Arc TK) mJ = ½|m Arc TM – m Arc TK| mJ = ½(m Arc MN – m Arc LK) mJ = ½|m Arc MN – m Arc LK| mJ = ½(m Arc TMS – m Arc TS) mJ = ½|m Arc TMS – m Arc TS| Find Method 1 if and Method 2 Answer: 98 Find if Answer: 138 and Find Answer: 55 if and Find Answer: 58 if and Find x. Theorem 10.14 Multiply each side by 2. Add x to each side. Subtract 124 from each side. Answer: 17 Find x. Answer: 111 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 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 Answer: 75 Find x. Multiply each side by 2. Add 40 to each side. Divide each side by 6. Answer: 25 Find x. Answer: 9 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