MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Apply Other Angle Relationships in Circles Essential Question: How do we find the measures of angles inside or outside a circle? M2 Unit 3: Day 5 Lesson 6.5 Monday, August 24, 2015

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. Daily Homework Quiz ANSWER 38 Find the value of x. 1. Daily Homework Quiz 2. ANSWER 56

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. 3. Find the value of x. ANSWER 44 Daily Homework Quiz 4. ANSWER x = 54; y = 20

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. 5. Find the value of each variable. ANSWER x = 5; y = 10 Daily Homework Quiz

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. ANSWER 58 ANSWER x = ( 360 – 120) = x = Solve for x. Warm Ups

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. ANSWER – x = (( 2x + 4) + 28) One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle. Solve for x. ANSWER 45º Warm Ups

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Theorem 6.13 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.

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. EXAMPLE Find angle and arc measures Line m is tangent to the circle. Find the measure of the red angle or arc. SOLUTION = 1212 (130 o ) = 65 o = 2 (125 o ) = 250 o b.m KJL a.m 1

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. GUIDED PRACTICE Find the indicated measure. SOLUTION = 1212 (210 o ) = 105 o m 1 Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. GUIDED PRACTICE Find the indicated measure. = 2 (98 o ) = 196 o m TSR SOLUTION Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. GUIDED PRACTICE Find the indicated measure. = 2 (80 o ) = 160 o m XY SOLUTION Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Theorem 6.14 Angle Inside the Circle If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arc intercepted by the angle and its vertical angle.

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. EXAMPLE 2 Find the value of x. SOLUTION The chords JL and KM intersect inside the circle. Use Theorem xoxo = 1212 (mJM + mLK) xoxo = 1212 (130 o o ) Substitute. xoxo = 143 Simplify. EXAMPLE

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. 4. Find the value of the variable. SOLUTION The chords AC and CD intersect inside the circle. Use Theorem 6.14 Substitute. Simplify. = 1212 (y o + 95 o ) 78 o = y 61 78° (mAB + mCD) = 1212 Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Theorem 6.15 Angle Outside the Circle

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Theorem 6.15 Secant and Tangent

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Theorem 6.15 Two Tangents

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. Theorem 6.15 Two Secants

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. EXAMPLE Find an angle measure outside a circle Find the value of x. SOLUTION Use Theorem Substitute. Simplify. The tangent CD and the secant CB intersect outside the circle. = 1212 (178 o – 76 o ) xoxo = 51 x m BCD (mAD – mBD) = 1212

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. = 40= 63 Find the value of x. EXAMPLE Find an angle measure outside a circle

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. GUIDED PRACTICE Find the value of the variable SOLUTION The tangent JF and the secant JG intersect outside the circle. Use Theorem Substitute. Simplify. = 1212 (a o – 44 o ) 30 o = 104 a m FJG (mFG – mKH) = 1212 Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. GUIDED PRACTICE 6. Find the value of the variable. SOLUTION Use Theorem Substitute o 1212 [(x o ) –(360 –x) o ] Solve for x. xoxo = 1212 m TQR (mTUR – mTR) Because QT and QR are tangents, QR RS and QT TS Also,TS SR and CA CA. So, QTS QRS by the Hypotenuse-Leg Congruence Theorem, and TQS RQS. Solve right QTS to find that m RQS 73.7°. Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. 7. Find the value of x. 50° = 83° – x x = 33° Guided Practice

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. EXAMPLE 4 Solve a real-world problem SCIENCE The Northern Lights are bright flashes of colored light between 50 and 200 miles above Earth. Suppose a flash occurs 150 miles above Earth. What is the measure of arc BD, the portion of Earth from which the flash is visible? (Earth’s radius is approximately 4000 miles.)

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. EXAMPLE 4 Solve a real-world problem SOLUTION Use Theorem Substitute. 149 o 1212 [(360 o – x o ) –x o ] Solve for x. xoxo 31 = 1212 m BCD (mDEB – mBD) Because CB and CD are tangents, CB AB and CD AD Also,BC DC and CA CA. So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA. Solve right CBA to find that m BCA 74.5°. ANSWER The measure of the arc from which the flash is visible is about 31 o.

MM2G3 Students will understand properties of circles. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 b Understand and use properties of central, inscribed, and related angles. MM2G3 d Justify measurements and relationships in circles and geometric and algebraic properties. MM2G3 Students will understand properties of circles. Homework Page 214 # 1 – 15 all