2. Measuring Arcs and Angles in a Circle SECONDARY LEVEL  Session #1  Presented by: Dr. Del Ferster

3.  First, we’ll look at some basic terminology dealing with circles. (Just in case you’re like I was as a HS teacher, and avoid geometry like it was the Plague!)  We’re going to spend some time examining the relationships between arcs and angles in a circle. ◦ We’ll focus on central angles ◦ We’ll focus on inscribed angles ◦ We’ll look at the tougher cases too:  Angles formed inside the circle, but not at center  Angles formed outside the circle.

4.  We’ll wrap it up with a few problems to work together.  As usual, I have lots of handouts that you can have.

5. How many of these do you recall?

6.  A circle is a locus of all points in a plane that are equidistant from a given point.  This fixed point is called the center of the circle  This is circle C

7.  Any segment with endpoints on the circle is a chord.  Chords can pass through the center but do not have to.  Chords are segments, and are usually named by their endpoints.

8.  Any chord that passes through the center is called a diameter  Segments and are radii. Radii is the plural of Radius ◦ A radius is half the length of the diameter

9.  A line that intersects the circle at exactly two points.  is a secant line

10.  A line that intersects a circle at exactly one point.  is a tangent line.

11.  A central angle has the center of the circle as its vertex.  Its sides are made up of two radii.

12.  An angle whose vertex is on a circle and whose sides are determined by two chords.  is an inscribed angle

13.  A portion of the circumference of the circle consisting of two points on the circle and all the points on the circle needed to connect them by a single path.  Example:

14.  An arc that lies in the interior of angle.  Example:

15.  Minor arc – Part of a circle that measures less than 180°  Major arc – Part of a circle that measures between 180° and 360°.  Semicircle – An arc whose endpoints are the endpoints of a diameter of the circle.  Note : major arcs and semicircles are named with three points and minor arcs are named with two points

17. Central angles and Inscribed angles

18.  The vertex of a central angle is at the center of the circle. is a central angle.  The sides of a central angle are radii of the circle. and )  The measure of the intercepted arc of a central angle is equal to the measure of the central angle. (Measured in degrees) A D B M V

19.  The vertex of an inscribed angle is a point on the circle.  The sides of an inscribed angle are chords of the circle.  The measure of an inscribed angle is half the measure of its intercepted arc. P Q R 64°

20. Name the following: Two central angles: _____________ Find ____________________ Find ___________________ Name one of the longest chords: _________ What fractional part of the circle is _______ A D B M V 65˚

21.  Name the inscribed angle for arc  What does this angle measure?  What does measure? A B C D F 42°

22.  Name the inscribed angle for.  What does measure? A B C D F 68° 76° 88°

23. Central angles and Inscribed angles

26. The vertex is on the circle, so think “inscribed like”

27.  If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

29. Line m is tangent to the circle. Find the measure of angle 1

31. THEOREM If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is ___________________ ___________________. ½ the measure of the intercepted arc Measure of angle 1 = _____ Measure of angle 2 = _____ Measures of Angles Formed by Lines Intersecting ON a Circle = ½ the measure of the intercepted arc.

32. Vertex inside the circle but not at center Vertex outside the circle

33. THEOREM If two chords intersect in the interior of a circle, then the measure of each angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Measure of angle 1 = _____ Measure of angle 2 = ______ Measures of Angles Formed by Chords Intersecting INSIDE a Circle = ½ the SUM of the Intercepted Arcs

34. THEOREM If a secant and a tangent, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs. Measures of Angles Formed by Secants and/or Tangents Intersecting OUTSIDE a Circle = ½ the DIFFERENCE of the Intercepted Arcs

35. Case I: Tangent and a Secant Case II: Two Tangents Case III: Two Secants

36. Mrs. McConaughy Geometry: Circles35 Example 1 Lines Intersecting ON a Circle: Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. m < 1 = ½ intercepted arc 130 = ½ intercepted arc 260 m < 1 = ½ (150) m <1 = _____ 75 260 = intercepted arc Measures of Angles Formed by Lines Intersecting ON a Circle = ½ the measure of the intercepted arc.

37. Mrs. McConaughy Geometry: Circles36 Example 2 Lines Intersecting INSIDE a Circle: Finding the Measure Angles Formed by Two Chords Find x. ½ (174 + 106) = X ½ (280) = X 140 = X 140 Measures of Angles Formed by Chords Intersecting INSIDE a Circle = ½ the SUM of the Intercepted Arcs

38. The measure of an angle formed equals ½ its intercepted arc. The measure of an angle formed equals ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle. The measure of an angle formed equals ½ the difference of the measures of the 2 arcs intercepted by the angle.

39.  Conor says that will be fun!

40. Grab your pens and pencils, and let’s do some math! Feel free to work by yourself or with a partner.

41. Solve for x if and if A B C

42. Two secants drawn to a circle from an external point intercept arcs that are 122° and 68°. Find the measure of P 122° 68°

43. A central angle intercepts an arc that is 5/12 of the circle. Find the measure of angle x. of circle O O x

44. Find the measure of angle x. x 92° 44°

45. AB & AC are tangent to the circle, and Find the measure of A B C D 104°

46. Find the value of x.

47.  Find the indicated measures in circle P. a) b)

48. Find the value of x.

51. ProblemAnswer 1 2 3 4 5

52. ProblemAnswer 6 7A 7B 8 9 10

54.  Next time we’ll be looking at:  1.Areas of circles, and sectors  2. Finding the area of a segment of a circle. ◦ We’ll use the traditional formula for area of a triangle ◦ We’ll also sneak a bit of trig into the mix.