Warm up Find the measure of each lettered angle.

UNIT OF STUDY Lesson 6.3 ARCS AND ANGLES TOPIC VII - CIRCLES UNIT OF STUDY Lesson 6.3 ARCS AND ANGLES

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

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.

Arcs and Angles Two types of angles in a circle Central Angles Angle whose vertex is at the center of a circle. O B  AOB is 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

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

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

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

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

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

Cyclic Quadrilaterals Arcs and Angles Cyclic Quadrilaterals Find each lettered measure 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°. 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°.

Cyclic Quadrilaterals Find each lettered measure.

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

Practice (FRI Skill development)

TOPIC VII - CIRCLES Arcs Length

Arcs LenGTH Arc Definition m COR = 100o CR = 100o 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. o R m COR = 100o A CR = 100o

Arcs LenGTH 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 360 or 30°. 12

Arcs LenGTH 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 the same!

The arc length is some fraction of the circumference of its circle. Arcs LenGTH The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4 The arc measure is half of the circle because 180° = 1 360° 2 The arc measure is one-third of the circle because 120° = 1 360° 3 The arc length is some fraction of the circumference of its circle.

Arcs LenGTH 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 the circle because 180° = 1 360° 2

Arcs LenGTH 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 AB = 18.84 m Circle O Length of CD= 180° 2 (4 in) 360° Or CD= 180° 8 in CD = 12.56 in

l = x . 2 r 3600 Arcs LenGTH Arc Length conjecture The length of an arc equals the measure of the arc divided by 360° times the circumference l = x . 2 r 3600

Arcs LenGTH 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.

Arcs LenGTH Example: