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Find the measure of each lettered angle.

Warm up. Find the measure of each lettered angle. UNIT OF STUDY Lesson 6.3 ARCS AND ANGLES. TOPIC VII - CIRCLES. Warm up = (FRI Starting Point). You will learn …. (FRI Ending Point ). To discover relationships between an inscribed angle of a circle and its intercepted arc.

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Find the measure of each lettered angle.

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  1. Warm up Find the measure of each lettered angle.

  2. UNIT OF STUDYLesson6.3ARCS AND ANGLES TOPIC VII - CIRCLES

  3. Warmup = (FRI Starting Point)

  4. Youwilllearn …. (FRI Ending Point) To discover relationships between an inscribed angle of a circle and its intercepted arc

  5. Content…. (FRE - Research) Complete your investigations, and as you see the power point presentation, on your notebook and/or your packet, write down the key concepts and complete the conjectures.

  6. Arcs and Angles Twotypes of angles in a circle Central Angles A Angle whose vertex is at the center of a circle. O B  AOBis a central angle of circle O D Inscribed Angle B Angle that has its vertex on the circle and its sides are chords. O A  ABC is an inscribed angles of circle O D

  7. Arcs and Angles ArcDefinition It is two points on the circle and the continuous (unbroken) part of the circle between the two points. A Minor arc is an arc that is smaller than a semicircle and are named by their end points B o Major arc is an arc that is larger than a semicircle and are named by their end points and a point on the arc C Example: AB Example: ABC

  8. Arcs and Angles InscribedAngleProperties Incribed Angle Conjecture C The measure of an angle inscribed in a circle is one-half (1/2) the measure of the intercepted arc. 100o R o 50o A m CAR = ½ m COR

  9. Arcs and Angles InscribedAngleInterceptingthesamearc Incribed Angle intercepting Conjecture A Inscribed angles that intercept the same arc are congruent 80o 80o P B Q  AQB   APB

  10. Arcs and Angles AnglesInscribed in a Semicircle Angle Inscribed in a semicircle Conjecture A 90o 90o Angles inscribed in a semicircle are right angles 90o B

  11. Arcs and Angles CyclicQuadrilaterals Cycle quadrilaterals Conjecture A 101o The opposite angles of a cyclic quadrilateral are supplementary 132o 48o 79o B

  12. Arcs and Angles CyclicQuadrilaterals Findeachletteredmeasure By the Cyclic Quadrilateral Conjecture, w +100° = 180°, so w = 80°.  PSR is an inscribed angle for PR. m PR = 47°+73° = 120°, so by the Inscribed Angle Conjecture, x = ½(120°) =60°. By the Cyclic Quadrilateral Conjecture, x + y = 180°. Substituting 60° for x and solving the equation gives y =120°. By the Inscribed Angle Conjecture, w = ½ (47° + z). Substituting 80° for w and solving the equation gives z = 113°.

  13. CyclicQuadrilaterals Find each lettered measure.

  14. Arcs and Angles ArcsbyParallellines secant Parallel lines intercepted Arcs Conjecture A D Parallel lines intercept congruent arcs on a circle. B C AD  BC

  15. Practice (FRI Skilldevelopment)

  16. ArcsLength TOPIC VII - CIRCLES

  17. ArcsLenGTH ArcDefinition It is two points on the circle and the continuous (unbroken) part of the circle between the two points. C Minor arc is an arc that is smaller than a semicircle and are named by their end points 100o The measure of the minor arc is the measure of the central angle. R o m COR = 100o A CR = 100o

  18. ArcsLenGTH The measure of the arc from 12:00 to 4:00 is equal to the measure of the angle formed by the hour and minute hands A circular clock is divided into 12 equal arcs, so the measure of each hour is 360or 30°. 12

  19. ArcsLenGTH Because the minute hand is longer, the tip of the minute hand must travel farther than the tip of the hour hand even though they both move 120° from 12:00 to 4:00. So the arc length is different even though the arc measure is thesame!

  20. ArcsLenGTH The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4 The arc measure is half of thecircle because180° = 1 360° 2 The arc measure is one-third of thecirclebecause120° = 1 360° 3 The arc length is some fraction of the circumference of its circle.

  21. ArcsLenGTH To find the arcs length we have to follow this steps Step 1: find what fraction of the circle each arc For AB and CED find what fraction of the circle each arc is The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4 The arc measure is half of thecirclebecause 180° = 1 360° 2

  22. ArcsLenGTH Step 2: Find the circumference of each circle Circle T C = 2(12 m) C= 24 m Circle O C= 2 (4 in.) C= 8  in Step 1: find what fraction of the circle the arc is Step 3: Combine the circumferences to find the length of the arcs Circle T Length of AB= 90° 2 (12m) 360° Or AB= 90° 24 m 360° AB = 18.84 m Circle O Length of CD= 180° 2 (4 in) 360° Or CD= 180° 8 in 360° CD = 12.56 in

  23. ArcsLenGTH ArcLengthconjecture The length of an arc equals the measure of the arc divided by 360° times the circumference l = x . 2 r 3600

  24. ArcsLenGTH Remember: The arc is part of a circle and its length is a part of the circumference of a circle. The measure of an arc is calculated in units of degrees, but arc length is calculated in units of distance (foot, meters, inches, centimeter.

  25. ArcsLenGTH Example:

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