LEARNING TARGETHOMEWORK Geometry I can examine the relationship between central and inscribed angles by applying theorems about their measure. I can solve the unknown measure of arcs and angles in a circle. Homework on page 607, #12 to #30 (Even Numbers Only) Arcs and Chords

REVIEW Identify the following parts of the circle. 1. DC 2. AB 3. AC 4. line E 5. DC D C A B E chord radius diameter tangent secant radiusdiameterchordsecanttangent Note: The following are possible answers. midpoint

Circles Chapter 10 Sections 10.1 –10.3

Types of Angles Central angle - the vertex is on the center. Inscribed angle - the vertex is on the circle.

Types of Arcs P M O Major arc Minor arc Semicircle N Example: MO Example: MNO or MN Example: MON - the measure is more than 180 ° - the measure is less than 180 ° - the measure is equal to 180 °

Solving Unknown Arcs and Angles On the next slides… you will use a white board (or a filler with printing paper), and a marker to solve and answer the given problems. you will be given 30 seconds to solve each of the math problems. at the end of each problem, you will raise your white board with your answer on it. Make sure you box your answer. Have fun!

Measure of Arcs & Angles If ∠ ABC is 80°, what is the measure of arc AC? In a circle, the measure of the central angle is always equal to the measure of its intercepted arc. x° A B ° C n° x = n m ∠ ABC = m AC m AC = 80°

Measure of Arcs & Angles SOLUTION: a. measure of minor arc m ∠ xyz = m xz (since ∠ xyz is a central angle) b. measure of major arc major arc = 360° – m xz (minor arc) 68° =360° – 68° m xz (major arc) = 292° 292° m xz = 68° EXAMPLE: In the diagram below, if the m ∠ xyz is 68°, find the measure of a.) minor arc and b.) major arc. x y z

Measure of Arcs & Angles n°n° x°x° A B C The measure of inscribed angle is always equal to ½ the measure of its intercepted arc. x = ½ n or 2x = n m ∠ ABC = ½ (m AC) If ∠ ABC is 35°, what is the measure of arc AC? m AC = 70°

Measure of Arcs & Angles SOLUTION: Inscribed angle = ½ (intercepted arc) ∠ ABC = ½ (68°) ∠ ABC= 34° 68° 34° A B C EXAMPLE: If the measure of the minor arc below is 68°, find the measure of the inscribe angle, ∠ ABC.

x 24 ° this is the arc BC D Centre of Circle SOLUTION: Angle x is a central angle. Therefore, ∠ x = arc BC. Arc BC is an intercepted arc of inscribe angle ABC. Since inscribed angle = ½(intercepted arc), therefore, the intercepted arc is twice the inscribe angle. n = 2x ∠ x = 2 (24) ∠ x = 48° PROBLEM: If angle BAC is 24°, solve for x A B C Examine the diagram and solve

105 ° x O Arc PN Centre of Circle SOLVE: If m ∠ PON is 105°, what is the measure of (a) arc PN? (b) m ∠ PMN? M N P Examine the diagram and solve SOLUTION: Angle PON is a central angle. Therefore, ∠ PON = arc PMN. arc PMN = 105° a. Arc PN = ? = 360° – 105° Arc PN = 255° b. ∠ PMN = ? ∠ PMN an inscribe angle and arc PN is its intercepted arc. Since inscribed angle = ½(intercepted arc), therefore, ∠ PMN = ½ (255°) ∠ PMN = 127.5°

Remember… In a circle, the measure of the central angle is always equal to the measure of its intercepted arc. x = n The measure of inscribed angle is always equal to ½ the measure of its intercepted arc. x = ½ n or 2x = n

EXIT SLIP Use the diagram below to answer the following question: 1.Find m BC 2.Find the m BDC 3.Find m ∠ BAC A B C D 50°