Find Arc Measures

Find Arc Measures Sunday, September 14, 2014 Essential Question: How do we use angle measures to find arc measures? Lesson 6.2 M2 Unit 3: Day 2

Find xand y. 1. x = 60; y = 60 ANSWER 2. x = 35; y = 35 ANSWER Warm Ups

Give the name that best describes the figure . 1. b. a. CD AB tangent secant ANSWER ANSWER FD EP c. d. chord radius ANSWER ANSWER Daily Homework Quiz

ANSWER One tangent; it is a vertical line through the point of tangency. Daily Homework Quiz Tell how many common tangents the circles have . 2.

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. 2 2 2 Daily Homework Quiz 3.

12 ANSWER ANSWER 12 Daily Homework Quiz 4. Find x.

Arcs in a stained glass window http://free-stainedglasspatterns.com/2curvesround.html

ARCS The part or portion on the circle from some point B to C is called an arc. B C A O Arcs : Example: B Semicircle: An arc that is equal to 180°. A Example: C

Minor Arc & Major Arc O A minor arc is an arc that is less than 180° Minor Arc : A minor arc is named using its endpoints with an “arc” above. A Example: Major Arc: A major arc is an arc that is greater than 180°. B B A major arc is named using its endpoints along with another point on the arc (in order). A Example: C

Example: ARCS Identify a minor arc, a major arc, and a semicircle, given that is a diameter. E D A C F Minor Arc: Major Arc: Semicircle: Lesson 8-1: Circle Terminology

Central Angles A is a central angle P B NOT A Central Angle (of a circle) Central Angle (of a circle) Central Angle (of a circle) A central angle is an angle whose vertex is the center of the circle and whose sides intersect the circle.

Measuring Arcs A P B The measure of an arc is the same as the measure of its associated central angle.

1 . TQ TQis a minor arc, so m TQ=120o. 2 . QRT QRTis a major arc, so m QRT=240o. GUIDED PRACTICE Identifying arcs Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. SOLUTION SOLUTION

3 TQRis a semicircle, so m TQR =180o. 4 . TQR . QS QSis a minor arc, so m QS =160o. GUIDED PRACTICE Identifying arcs Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. SOLUTION SOLUTION

5 TSis a minor arc, so m TS =80o. 6 . TS . RST RSTis a semicircle, so m RST =180o. Identifying arcs Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. SOLUTION SOLUTION

Find the measure of each arc of P, where RTis a diameter. 9. 7. RTS RST RS 8. 7. RSis a minor arc, so mRS=mRPS=110o. 8. – RTSis a major arc, so mRTS = 360o 110o = 250o. 9. RT is a diameter, so RSTis a semicircle, and mRST=180o. Find measures of arcs SOLUTION

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Arc Addition Postulate

Survey 10. = + 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. 10. mAC mAB mBC mAC Find measures of arcs SOLUTION = 29o + 108o = 137o

Survey 11. 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. mACD = mAC + mCD 11. mACD Find measures of arcs SOLUTION = 137o + 83o = 220o

Survey – mADC = 360o 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. 12. mAC 12. mADC EXAMPLE 2 Find measures of arcs SOLUTION = 360o – 137o = 223o

Survey 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. mEBD = 360o – mED 13. 13. mEBD EXAMPLE 2 Find measures of arcs SOLUTION = 360o – 61o = 299o

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.

14. 15. 14. CDEF because they are in the same circle and mCD = mEF 15. RSand TUhave the same measure, but are not congruent because they are arcs of circles that are not congruent. EXAMPLE 3 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

VXYZ because they are in congruent circles and mVX=mYZ . 16. 16. EXAMPLE 3 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

ABCD because they are in congruent circles and mAB=mCD . 17. GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

18. MNand PQhave the same measure, but are not congruent because they are arcs of circles that are not congruent. GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

Definition Review Central angle An angle whose vertex is the center of the circle. Semicircle An arc with endpoints that are the endpoints of a diameter. Arc An unbroken part of a circle. Part of a circle measuring less than 180° Minor arc Major Part of a circle measuring between 180° and 360°

Definition Review Measure of a minor arc The measure of its central angle. Measure of a major arc The difference between 360° and the measure of the related minor arc. Congruent Circles Two circles with the same radius Two arcs with the same measure and in the same circle or congruent circles. Congruent Arcs

Homework Page 195-196 # 1 -12 all, 15, # 19 – 24 all, 32 – 34 all.

Find Arc Measures