Geometry 9.3 Arcs and Central Angles Central Angles An angle with the vertex at the center of the circle. A AQX, AQB, and YQX are examples of central angles. 7 X Q Y B 7 7 Arc An unbroken part of the circle. A AB X Q B Y XBA Please put minor arc, major arc, and semicircle in the same box on your Vocab List!!! 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. A A AX X X Q B Y Major Arcs Q Has a measure between 180 and 360 degrees.Y Needs three letters in its symbol. A The measure of a minor arc is equal to the measure of its central angle. X Q Y B AXY B XBY Using the letters shown in the diagram, name: X W Q 1. four central angles 7 XQY YQZ 7 7 WQX XQZ Z 7 2. two semicircles WXY XYZ 3. four minor arcs WX YX ZY WZ 4. four major arcs WXZ Y WZX Are these the same? YZX ZXY Adjacent Arcs Arcs with exactly one point in common. J IJ and JK are adjacent arcs. I K Are arcs that overlap adjacent? No, because they would have more than one common point. Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs. B mBC + mCD = mBCD C A Find the mistake on your handout. D Minor arc only needs two letters. Find each measure. 45 5. PCQ 60o 8. SQ 120o 11. SPQ 240o 6. ST 45o 9. SCQ 120o 12. PT 135o 15. TSQ 14. SPT 97.5o 360 – 45 = 315o 7. SQP 180o 10. SCP 180o 13. TCP 135o T S 135o C 120o P Q 60 Find the measure of each numbered angle. O is the center of the circle. 17. 18. 1 1 60o 2 O 2 O O 1 m 1 = 180o – m 2 m 2 = 180o – m 1 7 120o 40 1 7 O 19. 7 240 7 16. 40o 140o Congruent Arcs Arcs in the same circle or congruent circles that have equal measures are congruent. A T R RY = QA ≠ SP Q X B S P Y C XY ~ = AB but neither arc is congruent to ST because circle P is not congruent to the other two circles. Theorem In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. J M 2 ~ LM. If m 1 = m 2, then JK = 7 7 ~ LM, then m 1 = m 2. If JK = 7 7 1 L K • The figure shows two concentric circles with center N. Classify each statement as true of false A 45 B V W AB VW 20. mBC 45 True 21. 22. mDNC 90 23. mXY 45 True 24. VW WX True N False X E False Z Y D 25. AED VZY False True/False: mAB = mVW True C HW P. 341-342 WE 1-11, 16-18 for 17-18 see example P. 340 Note-do constructions 8-10 during this chapter