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. Use Inscribed Angles and Polygons How do we use inscribed angles of circles? M2 Unit 3: Day 4 Lesson 6.4 Tuesday, October 20, 2015

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 1. Find the value of x in C. Explain.. ANSWER 6 ; If a diameter of a circle is to the chord, then the diameter bisects the chord and its arc. Daily Homework Quiz

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 2. Find the value of x in C. Explain.. ANSWER 4 ; In the same circle, if two chords are equidistant from the center, then they are. = ~ Daily Homework Quiz

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 3. Determine whether RS is a diameter. ANSWER Yes. Sample answer: RS is the bisector of TU by Theorem 5.3. Then RS is a diameter of the circle by Theorem Daily Homework Quiz

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. ANSWER The measure of the interior angles of a quadrilateral are 80º, 100º, 55º, and 5xº. Find the value of x. 2. Two supplementary angles have measures 6xº and 12xº. Find each angle measure. ANSWER 60º; 120º Warm Ups

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. ANSWER Solve 3x = ( 4x + 12) Solve 80 = ( 360 – 2x) Warm Ups

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 Inscribed Angle Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle) No! Yes! Examples: Inscribed Angle Intercepted Arc B A C

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. Theorem 6.9 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. B C A O

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 Use inscribed angles a. m T mQRb. Find the indicated measure in P. SOLUTION M T = mRS = (48 o ) = 24 o a. mQR = 180 o mTQ = 180 o 100 o = 80 o. So, mQR = 80 o. – – mTQ = 2m R = 2 50 o = 100 o. Because TQR is a semicircle, b.

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 measure of the red arc or angle. 1. SOLUTION M G = mHF = (90 o ) = 45 o a. Guided Practice

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 measure of the red arc or angle. 2. SOLUTION mTV = 2m U = 2 38 o = 76 o. b. Guided Practice

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. Theorem 6.10 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A B D C

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 measure of an intercepted arc Find mRS and m STR. What do you notice about STR and RUS ? SOLUTION From Theorem 6.9, you know that mRS = 2m RUS = 2 (31 o ) = 62 o. Also, m STR = mRS = (62 o ) = 31 o. So, STR RUS EXAMPLE

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 3 Standardized Test Practice SOLUTION Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem 6.10 Also, KJL and KML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles. So, by Theorem 10.8, the correct answer is C.

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 measure of the red arc or angle. 3. SOLUTION ZYNZXN 72° Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem Also, KJL and KML intercept the same arc, so they must also be congruent. ZXN Guided Practice

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 mLJM. mLJM = 5(3.5) – 7 = 10.5 Substitute 3.5 for b. 5b – 7 = 3b Substitute the given values. 2b – 7 = 0 Subtract 3b from both sides. 2b = 7 Add 7 to both sides. b = 3.5 Divide both sides by 2. mLJM = mLKM m  LJM and m  LKM both intercept LM. Guided Practice

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 mEDF. 2x + 3 = 75 – 2x Substitute the given values. 4x = 72 Add 2x and subtract 3 from both sides. x = 18 Divide both sides by 4. mEDF = 2(18) + 3 = 39° mEDF = mEGF m  EGF and m  EDF both intercept EF. Guided Practice

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. 18 An angle inscribed in a semicircle is a right angle. R P 180  S 90  Theorem 6.11 Inscribed Right Angle Theorem

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. Find a. WZY is a right angle WZY is inscribed in a semicircle. mWZY = 90 Def of rt.  5a + 20 = 90 Substitute 5a + 20 for m  WZY. 5a = 70 Subtract 20 from both sides. a = 14 Divide both sides by 5. EXAMPLE

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. 6. Find z. 8z – 6 = 90 Substitute. 8z = 96 Add 6 to both sides. z = 12 Divide both sides by 8. ABC is a right angle ABC is inscribed in a semicircle. mABC = 90 Def of rt.  Guided Practice

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. 21 m  DAB + m  DCB = 180  m  ADC + m  ABC = 180  If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. Theorem 6.12 Inscribed Quadrilateral Theorem

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 5 Find the value of each variable. a. SOLUTION PQRS is inscribed in a circle, so opposite angles are supplementary. a. m P + m R = 180 o 75 o + y o = 180 o y = 105 m Q + m S = 180 o 80 o + x o = 180 o x = 100 EXAMPLE

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 5 b. JKLM is inscribed in a circle, so opposite angles are supplementary. m J + m L = 180 o 2a o + 2a o = 180 o a = 45 m K + m M = 180 o 4b o + 2b o = 180 o b = 30 4a = 180 6b = 180 Find the value of each variable. b. SOLUTION EXAMPLE

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 7. Find the value of each variable. SOLUTION ABCD is inscribed in a circle, so opposite angles are supplementary. m A + m C = 180 o y°+ 68 o = 180 o y = 112 m D+ m B = 180 o 82 o + x o = 180 o x = 98 Guided Practice

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 8. Find the value of each variable. SOLUTION STUV is inscribed in a circle, so opposite angles are supplementary. m S + m U = 180 o c°+ (2c – 6 ) o = 180 o c = 62 m V+ m T = 180 o 8x o + 10x o = 180 o x = 10 (3c) o = 186 o 18x o = 180 o Guided Practice

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. 9. Find the angle measures of GHJK. mG + mJ = 180 GHJK is inscribed in a . 3b b + 20 = 180 Substitute the given values. 9b + 45 = 180 Simplify. 9b = 135 Subtract 45 from both sides. b = 15 Divide both sides by 9. mG = 3(15) + 25 = 70 Substitute 15 for b mJ = 6(15) + 20 = 110 in each expression. mK = 10(15) – 69 = 81 mH + mK = 180  H and  K are supp. mH + 81 = 180 Substitute 81 for m  K. mH = 99 Subtract 81 from both sides Guided Practice

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. Step 2 Find the measure of each angle. Example 4 Continued mG = 3(15) + 25 = 70 Substitute 15 for b mJ = 6(15) + 20 = 110 in each expression. mK = 10(15) – 69 = 81 mH + mK = 180 H and  K are supp. mH + 81 = 180 Substitute 81 for m  K. mH = 99 Subtract 81 from both sides

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. 35 o xoxo yoyo Q L K J Find x and y. 90 o Guided Practice 70 o 110 o 180 o

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: pg (#1, 2, 5, 6, 8-18, 22)