Check.4.40 Find angle measures, intercepted arc measures, and segment lengths formed by radii, chords, secants, and tangents intersecting inside and outside circles. CLE Develop the role of circles in geometry, including angle measurement, properties as a geometric figure, and aspects relating to the coordinate plane. Spi.4.13 Identify, analyze and/or use basic properties and theorems of circles to solve problems (including those relating right triangles and circles) Secants, Tangents, and Angle Measures

A line that intersects a circle in two points is called a secant. An education isn't how much you have committed to memory, or even how much you know. It's being able to differentiate between what you know and what you don't. Anatole France. If a secant and a tangent intersect at the point of tangency then the measure of each angle is ½ the measure of the intercepted arc. If two secants intersect in the interior of a circle, then the measure of the angle formed is ½ the measure of the arcs formed by the angle and its vertical angle.

Prove the theorem StatementReason Secant RT and SUGiven Given: Secant RT and SU Prove :m  1 = ½ (mST and mRU) StatementReason Secant RT and SUGiven m  1 = m  2 + m  3 Exterior Angle Theorem StatementReason Secant RT and SUGiven m  1 = m  2 + m  3 Exterior Angle Theorem m  2 =½mST, m  3=½mRU Measure of inscribed  = ½ the measure of the arc StatementReason Secant RT and SUGiven m  1 = m  2 + m  3 Exterior Angle Theorem m  2 =½mST, m  3=½mRU Measure of inscribed  = ½ the measure of the arc m  1 = ½ (mST + mRU) Substitution

Secant – Secant Angle Find m  2 if measure arc BC=30 and measure of arc AD = 20 m  1= ½ (arc AD + arc BC) = ½ (30+20) = 25 m  2 = m  1 = = 155 Or m  2= ½ (arc AB + arc DC) arc AB + arc DC + arc AD + arc BC = 360 arc AB + arc DC = 360 – 50 = 310 ½ (310) = 155

Find m  ABC if measure of arc AB = 102  m  ABC = ½ (measure arc ADB) arc ADB = 360 – 102 = 258 m  ABC = ½ (258) = 129 m  RPS = ½ (measure arc RPS) arc PS = = 110 m  STP = ½ (110) = 55 Find m  RPS if measure of arc PT= 114  and measure of arc TS = 136 

Secant Secant Angle X = ½ (mAE – m DB) X= ½ (120 – 50) X = ½ (70) = = ½ (141 – x) 124= x X = = 17

m  S = ½ (mPQR – mPR) Let mPR = x 11= ½ (360 – x –x) 11 = ½ (360 – 2x) 11 = 180 – x x = 169

40= ½ (x –(360-x)) 40 = ½ ( x) 40 = x x = 220 X

Secants, Tangents, and Angle Measures Summary Practice Assignment Page 732, even* Honors Page 732, even*, 34,38 If two secants intersect in the interior of a circle, then the measure is ½ the measure of the arcs If a secant and a tangent intersect at the point of tangency then the measure of each angle is ½ the measure of the intercepted arc.