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Geometry

Geometry. 9.3 Arcs and Central Angles. A. X. B. Q. Y. Central Angles. An angle with the vertex at the center of the circle. AQX, AQB, and YQX are examples of central angles. 7. 7. 7. A. X. B. Q. Y. Arc. An unbroken part of the circle. AB. XBA. A. A. A. X. X. X. B.

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Geometry

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  1. Geometry 9.3 Arcs and Central Angles

  2. A X B Q Y Central Angles • An angle with the vertex at the center of the circle. AQX, AQB, and YQX are examples of central angles. 7 7 7

  3. A X B Q Y Arc • An unbroken part of the circle. AB XBA

  4. A A A X X X B B B Q Q Q Y Y Y Measures of an arc Semicircle Minor Arcs Has a measure of 180 degrees. Needs three letters in its symbol. Has a measure between 0 and 180 degrees. Needs only two letters in its symbol. AX Major Arcs Has a measure between 180 and 360 degrees. Needs three letters in its symbol. XBY The measure of a minor arc is equal to the measure of its central angle. AXY

  5. X W Q Y Z 7 WQX XQY YQZ XQZ 7 7 7 WXY XYZ WX YX ZY WZ WXZ WZX YZX ZXY Are these the same?

  6. J I K Adjacent Arcs • Arcs with exactly one point in common. IJ and JK are adjacent arcs. Are arcs that overlap adjacent? No, because they would have more than one common point.

  7. Arc Addition Postulate • The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs. B C mBC + mCD = mBCD A D Find the mistake on your handout. Minor arc only needs two letters.

  8. T S C P Q Find each measure. 6. ST 9. 12. PT 15. 7. SQP 10. 13. 5. 8. SQ 11. SPQ 14. SPT 135o 45o 180o 60o 120o 120o 120o 180o 135o 135o 240o 97.5o 360 – 45 = 315o

  9. 1 2 1 O O O 2 O 1 1 Find the measure of each numbered angle. O is the center of the circle. 60o 140o 120o m 1 = 180o – m 2 7 7 40o 7 m 2 = 180o – m 1 7

  10. A T R S P Q B Y X C Congruent Arcs • Arcs in the same circle or congruent circles that have equal measures are congruent. RY = QA ≠ SP ~ XY = AB but neither arc is congruent to ST because circle P is not congruent to the other two circles.

  11. J M 1 K 2 L Theorem • In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. ~ 7 7 If m 1 = m 2, then JK = LM. ~ 7 7 If JK = LM, thenm 1 = m 2.

  12. A B V W N C X E Z Y D • The figure shows two concentric circles with center N. Classify each statement as true of false 20. 21. 22. 23. 24. 25. True False False True True False True/False: mAB = mVW True

  13. HW • P. 341-342 WE 1-11, 16-18 for 17-18 see example P. 340

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