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Understanding Arc Measures in Circles: Minor, Major, and Semicircles

This educational resource explains how to measure arcs in circles, focusing on minor arcs, major arcs, and semicircles. Through various examples, it illustrates how to compute arc measures based on given angles and surveys, including how to identify congruent arcs and understand their properties. The content also includes guided practice problems that reinforce the learning objectives. Students will gain confidence in applying their knowledge of arc measures and improve their problem-solving skills in geometry.

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Understanding Arc Measures in Circles: Minor, Major, and Semicircles

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

  2. Find the measure of each arc of P, where RTis a diameter. c. a. RTS RST RS b. a. RSis a minor arc, so mRS=mRPS=110o. b. – RTSis a major arc, so mRTS = 360o 110o = 250o. c. RT is a diameter, so RSTis a semicircle, and mRST=180o. EXAMPLE 1 Find measures of arcs SOLUTION

  3. Survey a. = + 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. a. mAC mAB mBC mAC EXAMPLE 2 Find measures of arcs SOLUTION = 29o + 108o = 137o

  4. Survey b. 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 b. mACD EXAMPLE 2 Find measures of arcs SOLUTION = 137o + 83o = 220o

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

  6. 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 d. d. mEBD EXAMPLE 2 Find measures of arcs SOLUTION = 360o – 61o = 299o

  7. for Examples 1 and 2 GUIDED PRACTICE Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc.

  8. 1 . TQ TQis a minor arc, so m TQ=120o. 2 . QRT QRTis a major arc, so m QRT=240o. for Examples 1 and 2 GUIDED PRACTICE SOLUTION SOLUTION

  9. 3 TQRis a semicircle, so m TQR =180o. 4 . TQR . QS QSis a minor arc, so m QS =160o. for Examples 1 and 2 GUIDED PRACTICE SOLUTION SOLUTION

  10. 5 TSis a minor arc, so m TS =80o. 6 . TS . RST RSTis a semicircle, so m RST =180o. for Examples 1 and 2 GUIDED PRACTICE SOLUTION SOLUTION

  11. a. b. a. CDEF because they are in the same circle and mCD = mEF b. 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

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

  13. ABCD because they are in congruent circles and mAB=mCD . 7. for Example 3 GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

  14. 8. MNand PQhave the same measure, but are not congruent because they are arcs of circles that are not congruent. for Example 3 GUIDED PRACTICE Tell whether the red arcs are congruent. Explain why or why not. SOLUTION

  15. Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 1. BC o ANSWER minor arc, 32 Daily Homework Quiz

  16. CBE o ANSWER major arc, 212 Daily Homework Quiz Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 2.

  17. BCE o ANSWER semicircle, 180 Daily Homework Quiz Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 3.

  18. ~ ~ ~ = = = AE BC BC BC AE AE ANSWER mAFE = m BFC because the angles are vertical angles, so AFEBFC.Then arcs and are arcs that have the same measure in the same circle. By definition . Daily Homework Quiz Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. . 4. Explain why

  19. . o Two diameters ofPareABandCD.If m = 50 , find m and m . AC AD ACD o o ANSWER 310 ; 130 Daily Homework Quiz 5.

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