Lesson 9.5 Lines and Circles pp. 394-398

Objectives: 1. To find the measures of angles formed by intersecting lines based on the measures of the intercepted arcs. 2. To prove the relationships for such intersecting lines.

Theorem 9.17a The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

If mCD = 100 and mBE = 30, then mBAE = O If mCD = 100 and mBE = 30, then mBAE = 35°

Theorem 9.18 The measure of an angle formed by two secants that intersect in the interior of a circle is one-half the sum of the measures of the intercepted arcs.

If mAB = 40 and mDC = 30, then m1 = Q 1 If mAB = 40 and mDC = 30, then m1 = 35°

Theorem 9.19 The measure of an angle formed by a tangent and a secant that intersect at the point of tangency is one-half the measure of the intercepted arc: mHIJ = ½mHI.

P H I J 1 If mHI = 120, then mHIJ = 60°

EXAMPLE Find the measure of ABC if BC is tangent to ⊙Q at B and AB is a secant. mAB = 170. Answer A Q B C 1 mABC = mAB 1 2 mABC = (170) 1 2 mABC = 85

Theorem 9.17b The measure of an angle formed by a secant and a tangent that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

Theorem 9.17c The measure of an angle formed by the intersection of two tangents is one-half the difference of the measures of the intercepted arcs.

x 65º 111º x = 92º Theorem 9.18

x 64º 29º x = 122º Theorem 9.17a

x 23º 41º x = 9º Theorem 9.17a

58º x x = 244º Theorem 9.19

x 78º x = 102º Theorem 9.17c

67 x 128 x = 49º Theorem 9.17b

Homework pp. 396-398

►A. Exercises Find x. 9. x 86 36°

►A. Exercises Find x. 13. x 116°

18. Given: AC and BD are secants that 18. Given: AC and BD are secants that intersect inside circle E and form 1. Prove: m1 = ½(mAB + mDC) A D 1 B C E

■ Cumulative Review 24. Name the type of quadrilateral. Use the quadrilateral shown for the following questions. 24. Name the type of quadrilateral.

■ Cumulative Review 25. How do the diagonals relate? Use the quadrilateral shown for the following questions. 25. How do the diagonals relate?

■ Cumulative Review 26. perimeter. Use the quadrilateral shown for the following questions. If the diagonals of the quadrilateral shown are 10 and 6 inches respectively, give the 26. perimeter.

■ Cumulative Review 27. area. Use the quadrilateral shown for the following questions. If the diagonals of the quadrilateral shown are 10 and 6 inches respectively, give the 27. area.

■ Cumulative Review Use the quadrilateral shown for the following questions. 28. Name the type of polyhedron formed by using the quadrilateral shown as the base of a pyramid.