CIRCLES AND ANGLES Section 10-4, 10-6 spi.3.3.A, spi.3.3.B Jim Smith JCHS

Central Angles 1 = 63° 63° Central Angles Are Equal To The Measure Of The Intercepted Arc 63° 1 = 63° 1

The Vertex Of Inscribed Angles Are On The Circle And The Sides Are Contained In Chords. 2 4 3 1

The Measure Of An Inscribed Angle Is Equal To ½ The Measure Of The Intercepted Arc 1 2 4 3 2 = ½ (180 ) 2 = 90° 1 = ½ ( 100 ) 1 = 50° 100°

Any Angle With It’s Vertex On The Circle Is Equal To ½ Of The Arc. A secant and a tangent 2 secants

½ The Sum Of The Arcs If The Vertex Is Somewhere Inside The Circle But Not At The Center, The Angle Is Equal To ½ The Sum Of The Arcs 150° 8 = ½ (150 + 80) 8 = ½ ( 230 ) 8 = 115° 8 80°

½ The Difference Of The Arcs. If The Vertex Is Outside (A Distance Away From) The Circle, The Arc Is Equal To ½ The Difference Of The Arcs. 5 50° 5 = ½ (120 - 50) 5 = ½ ( 70 ) 5 = 35° 120°

= arc =1/2 arc 1 4 3 =1/2 (difference =1/2 (sum of of 2 arcs) 2 arcs)

D A C F E B AF is a tangent AC is a diameter CD = 100° CB = AE BE = 50° D A 1 2 4 C Find the Measures of the angles 5 F 3 E B

D A C F E B 50° AF is a tangent AC is a diameter CD = 100° 80° CB = AE BE = 50° D 80° 100° A 1 2 4 65° C 5 F 130/2 =65 3 E 65° B 50°

D 80° 1 = 80° 100° A 1 2 4 65° C 5 F 3 E 65° B 50°

D 80° 2 =1/2 (65) 2 = 32 ½ ° 100° A 1 2 4 65° C 5 F 3 E 65° B 50°

D A F C E B 50° 80° 100° 3 =1/2 (180+50) 3 =1/2 (230) 3 = 115° 65° 65° 4 65° F C 5 3 E 65° B 50°

D A C F E B 50° 4 =1/2 (65+50) 80° 4 =1/2 (115) 100° 4 = 57 ½ ° 65° 3 E 65° B 50°

D A C F E B 50° 5 =1/2 (180-65) 80° 5 =1/2 (115) 100° 5 =57 1/2° 65° 4 65° C 5 F 3 E 65° B 50°