Angle Relationships in Circles
Objectives: Have students discover three theorems
concerning relationships of different angles of circles.
Lesson:
Have students discover the following theorems:
Theorem 1: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. m<DAB = (1/2)mACB and m<EAB = (1/2)mAB 

Theorem 2: If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. m<AEB = (1/2)(mAB + mCD) m< AED = (1/2)(mAD + mCB) 

Theorem 3: If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 

m<ADC = (1/2)(mCA  mBC)  
m<EFG = (1/2)(mGHE  mEG)  
m<IKM = (1/2)(mMI  mJL) 
Here are the interactive GSP sketches of these theorems: Theorem 1, Theorem 2, Theorem 3.
Conclusion: Discuss the activity with the students and
have them describe the conclusions they were able to come to based upon
the activity. Review the theorems that the students should have
discovered during their investigations.
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