What’s Up for Today? LEARNING TARGET IN-CLASS HOMEWORK Geometry I can examine relationship between central and inscribe angles by applying theorems about their measure. I can solve the unknown measure of arcs and angles in a circle. Review on parts of circle Powerpoint presentation on circles Use classwork notes to show answers for math problems. Sage and scribe activity (hand out) Exit slips No homework

REVIEW Identify the following parts of the circle. A B E DC AB AC line E chord radius diameter tangent secant Note: The following are possible answers. radius diameter chord midpoint secant tangent

Circles Chapter 10 Sections 10.1 –10.3

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

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

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

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

Measure of Arcs & Angles 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) C n° If ∠ ABC is 35°, what is the measure of arc AC? m AC = 70° A x° B

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

Examine the diagram and solve PROBLEM: If angle BAC is 24°, solve for x 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° A D Centre of Circle 24 ° this is the arc BC x C B

Examine the diagram and solve SOLVE: If m ∠PON is 105°, what is the measure of (a) arc PN? (b) m ∠ PMN? 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° Centre of Circle M x N P 105° O Arc PN

Sage and Scribe Activity Students will work in pairs. The first partner, the “sage”, will talk about the math problem, while the other student, the scribe will write it. If the sage is correct, the scribe praises the sage. Otherwise, the scribe coaches, then praises. Students swap roles for the next problem. Students will work for 4 minutes at voice level 1 (whisper). If you are done with 4 math problems, you are encourage to try the challenge problem at the bottom of hand out #3. Have fun! 

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: Find m BC Find the m BDC Find m ∠BAC B D 50° A C