1) What is the Triangle Exterior Angle Theorem? 2) Solve for x. 3) Solve for y. (5x)° R Q T S A B C D (2y - 3)° (y - 1)°

Secant is a line that intersects a circle at two points. B A Secant is a line that intersects a circle at two points. F E

Theorem If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. A B C D 1 E Chords AD and BC intersect at E. m<1 = ½ (mAB +mCD)

Find each angle measure. 1. Find m<SQR Examples Find each angle measure. 1. Find m<SQR 2. Find m<ABE A B C D E 65° 37° P S Q T R 100° 32° 3. Find mMQ M R Q P N 160° 255°

You try Find m<AEB 5. Find mRS 6. Find m<LPO R S N L T 90° 100° C 139° 113° L O P M N 65° 6. Find m<LPO 80°

Theorem If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. Two Tangents Tangent and Secant Two Secants E F G H 2 C B D A 1 L K M N J 3 m<1 = ½ (mAD – mBD) m<2 = ½ (mEHG – mEG) m<3 = ½ (mJN – mKM)

Examples 1. 2. 3. E F G x° 132° P Q S R x° 174° 98° L K M N J x° 83° 25° 3.

P Q S R x° 200° 74° 4. E F G x° 225° 5. L K M N J x° 30° 25° 6.

Summing Up Angles in Circles Where’s the Vertex? What’s the Angle Measure? Examples 2 200° m<2=100° Vertex on a circle 1 120° m<1=60° Half of the arc Vertex inside a circle Half of the arcs added 44° 1 m<1= ½ (44°+86°) = 65° 86° 1 78° 202° 2 45° 125° Vertex outside a circle Half of the arcs subtracted m<2= ½ (125°-45°) = 40° m<1= ½ (202°-78°) = 62°

Whiteboards Practice 12-4 Workbook P 491 #1-12 all

Warm Up 1. Find mAF 2. Find LP 3. Find mYZ mLR= 100° L M N S R P Q 80° 26° mLR= 100° A D E F B C 160° 110° 48° On the white boards! 3. Find mYZ 68° 113° 49° X Y Z W

Last 4 m<FGJ m<HJK Solve for x 4. mCE E F G 31° 52° H J 130° K D 150° A B C D E G H J 25° 48° 83° Common mistakes: #1) Said 52, assumed that G was center Said 10.5, subtracted arcs instead of adding them #2) Said 90 instead of 65, assuming that HJ was diameter Said 130 instead of 65, assuming central angle

P 681 # 5 – 24 all, 36 P 691 # 2-14 even, 20- 24 even Period 1 Due Tuesday Period 2/4 Due Wednesday Exam on Thurs/Fri