Relationships of Angles Day 1



Objectives:  Have the students discover four theorems about angles in circles.

Lesson:  Have the students perform some investigations in GSP to help them deduce some theorems about relationships between angles of circles.  These theorems are as follows:


Theorem 1: If an angle is inscribed in a circle, then it's measure is half the measure of its intercepted arc.

m<ACE = 1/2m<ABC

Theorem 2: If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

m<ABC = m<ADC

Theorem 3: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

m<ABC = 90o if and only if AC is the diameter of the circle

Theorem 4: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

m<ABC + m<CDA = 180o

m<BCD + m<DAB = 180o

 

For the GSP activity click here. Here are the interactive GSP sketches for each of the theorems: Theorem 1, Theorem 2, Theorem 3, Theorem 4.


Conclusion:  Have the students discuss the conclusions they were able to form based on the activity.  Discuss the theorems that the students should have found during the activity.

 

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