section8

CIRCLES AND ANGLES.

Now we are going to discuss the possible relationships between circles and angles. We start with the following definition.
An angle ABC is called an angle inscribed in a circle O if the vertex B is a point on the circle O and segments AB and AC are chords of O. An angle OAC is called a central angle of a circle O if A and C are both points on the circle O.

Theorem 8.1 Suppose that O is a circle and an angle ABC is an angle inscribed in O. Then

Proof We need to show that an inscribed angle is equal to the half of the corresponding central angle. In order to do this we have to consider all possible cases that may happen for an inscribed angle.
Case 1: One of the angle sides passes through O.

Let's denote the degree measure of ABC as x. Then we need to show that the degree measure of AOC is 2x. Indeed, the triangle AOB is an isosceles one, therefore the degree measure of BAO is x. From the theorem regarding the sum of angles in a triangle it follows that
But angles BOA and AOC are supplementary angles, so

Case 2: O is between angle sides.

Let's denote the degree measure of ABC as x again. And let's denote the degree measure of OBC as y. Since the triangle BOC is an isosceles one and from the theorem regarding the sum of angles in a triangle it follows that
By the same reasoning we have that

Case 3: both sides of the angle are on the same side from O.

If we denote the degree measure of ABC as x and that of CBO as y then the degree measure of ABO is x+y. Following the same pattern of reasoning as the one for the previous two cases we have that

Corollary 1 Inscribed angles with sides passing through fixed points A and C and vertices located in the same half-plane with respect to the chord AC, all have the same degree measure.

Corollary 2 The degree measure of an inscribed angle ABC such that AC is a diameter of the circle O, is equal to 90 degrees.

We will not provide the proofs for the corollaries that follow immediately from the theorem 8.1.

EXERCISE SET #8

Problem 1. Consider a picture below:

Given that the degree measure of the inscribed angle ABC is equal to 60 degrees find the degree measure of the inscribed angle ADC.

Problem 2. Suppose that there is a circle (O,r). Let AB and CD be two distinct chords of (O,r) such that AB = CD = r. Prove that inscribed angles AXB and CYD where X and Y are two arbitrary points of (O,r) are equal. Hint: find the degree measures of AXB and CYD.