1. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Find Arc Measures Essential Question: How do we use angle measures to find arc measures? M2 Unit 3: Day 2 Tuesday, November 10, 2015 Lesson 6.2

2. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. ANSWER x = 60; y = 60 ANSWER x = 35; y = 35 1. Find x and y. 2. Warm Ups

3. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Daily Homework Quiz ANSWER secant 1. Give the name that best describes the figure. a.CD b.AB c. FD d.EP ANSWER tangent ANSWER chord ANSWER radius

4. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 2. Tell how many common tangents the circles have. ANSWER One tangent; it is a vertical line through the point of tangency. Daily Homework Quiz

5. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 3. Is AB tangent to C? Explain.. ANSWER Yes; 16 + 30 = 1156 = 34 so AB AC, and a line to a radius at its endpoint is tangent to the circle. 222 Daily Homework Quiz

6. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 4. Find x. ANSWER 12 ANSWER 12 Daily Homework Quiz

7. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. http://free-stainedglasspatterns.com/2curvesround.html Arcs in a stained glass window

8. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 8 ARCS The part or portion on the circle from some point B to C is called an arc. A B C Arcs : Semicircle: An arc that is equal to 180°. Example: O A B C

9. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 9 Minor Arc & Major Arc Minor Arc : A minor arc is an arc that is less than 180° A minor arc is named using its endpoints with an “arc” above. A B Example: Major Arc: A major arc is an arc that is greater than 180°. A major arc is named using its endpoints along with another point on the arc (in order). A B C Example: O

10. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Lesson 8-1: Circle Terminology10 Example: ARCS Identify a minor arc, a major arc, and a semicircle, given that is a diameter. A C D E F Minor Arc: Major Arc: Semicircle:

11. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Central Angles A central angle is an angle whose vertex is the center of the circle and whose sides intersect the circle. A P B is a central angle Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle)

12. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Measuring Arcs The measure of an arc is the same as the measure of its associated central angle. A P B

13. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. GUIDED PRACTICE Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. Identifying arcs TQ is a minor arc, so m TQ = 120 o. 1. TQ SOLUTION. QRT2 SOLUTION QRT is a major arc, so m QRT= 240 o.

14. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. GUIDED PRACTICE. TQR3 SOLUTION TQR is a semicircle, so m TQR = 180 o.. QS4 SOLUTION QS is a minor arc, so m QS = 160 o. Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. Identifying arcs

15. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.. TS5 SOLUTION TS is a minor arc, so m TS = 80 o.. RST6 SOLUTION RST is a semicircle, so m RST = 180 o. Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. Identifying arcs

16. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Find measures of arcs RS 7. RTS 8. RST 9. SOLUTION RS is a minor arc, so mRS = m RPS = 110 o. 7. RTS is a major arc, so mRTS = 360 o 110 o = 250 o. 8. – Find the measure of each arc of P, where RT is a diameter. 9. RT is a diameter, so RST is a semicircle, and mRST = 180 o.

17. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Arc Addition Postulate

18. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Find measures of arcs A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. Survey 10. mAC SOLUTION 10. mAC mAB =+ mBC = 29 o + 108 o = 137 o

19. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 11. mACD = mAC + mCD = 137 o + 83 o = 220 o A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. Survey SOLUTION 11. mACD Find measures of arcs

20. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 2 mADC mAC = 360 o – 12. = 360 o – 137 o = 223 o A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. Survey SOLUTION 12. mADC Find measures of arcs

21. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 2 13. mEBD = 360 o – mED = 360 o – 61 o = 299 o 13. mEBD A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. Survey SOLUTION Find measures of arcs

22. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Congruent arcs are two arcs with the same measure that are arcs of the same circle or of congruent circles. Congruent Arc Definition Congruent Circles Definition Congruent circles are two circles with the same radius.

23. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 3 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. 14. 15. SOLUTION 14. CD EF because they are in the same circle and mCD = mEF 15. RS and TU have the same measure, but are not congruent because they are arcs of circles that are not congruent.

24. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 3 Identify congruent arcs 16. VX YZ because they are in congruent circles and mVX = mYZ. Tell whether the red arcs are congruent. Explain why or why not. 16. SOLUTION

25. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. 17. AB CD because they are in congruent circles and mAB = mCD. SOLUTION

26. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. 18. SOLUTION MN and PQ have the same measure, but are not congruent because they are arcs of circles that are not congruent.

27. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. An angle whose vertex is the center of the circle. An arc with endpoints that are the endpoints of a diameter. An unbroken part of a circle. Part of a circle measuring less than 180° Part of a circle measuring between 180° and 360° Definition Review Central angle Semicircle Arc Minor arc Major

28. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. The measure of its central angle. The difference between 360° and the measure of the related minor arc. Two circles with the same radius Two arcs with the same measure and in the same circle or congruent circles. Definition Review Measure of a minor arc Measure of a major arc Congruent Circles Congruent Arcs

29. MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Homework Page 195-196 # 1 -12 all, 15, # 19 – 24 all, 32 – 34 all.