The Inscribed Angle Theorem

Introduction

An inscribed angle is formed when two secant lines intersect on a circle. It can also be formed using a secant line and a tangent line intersecting on a circle. A central angle, on the other hand, is an angle whose vertex is the center of the circle and whose sides pass through a pair of points on the circle, therefore subtending an arc. In this post, we explore the relationship between inscribed angles and central angles having the same subtended arc. The angle of the subtended arc is the same as the measure of the central angle (by definition).

inscribed-angle-theorem

In the first circle, \angle O is a central angle subtended by arc AB. Angle Q is an inscribed angle subtended by arc PR. In the second circle, \angle T is an inscribed angle and \angle V is a central angle. Both angles are subtending arc SU.

What can you say about the two angles subtending the same arc? Draw several cases of central angles and inscribed angles subtending the same arc and measure them. Use a dynamic geometry software if necessary.  Are your observations the same?

In the discussion below, we prove one of the three cases of the relationship between a central angle and an inscribed angle subtending the same arc.

Theorem

The measure of an angle inscribed in a circle is half the measure of the arc it intercepts. Note that this is equivalent to the measure of the inscribed angle is half the measure of the central angle if they intercept the same arc.

Proof

Let T be an inscribed angle and V be a central angle both subtending arc SU as shown in the figure. Draw line TV. This forms two isosceles triangles SVT and TVU since two of their sides are radii of the circle.

central angle theorem

In triangle SVT, if we let the measure of \angle STV be x, then angle TSV is also x. By the exterior angle theorem, the measure of angle SVW = 2x. This is also similar to triangle TVU.  If we let angle VTU = y, it follows that \angle WVU is equal to 2y.  In effect, the measure of the inscribed angle STU = x + y and the measure of central angle SVU = 2x + 2y = 2(x + y) which is what we want to prove.

inscribed-angle-case2and3

The proofs of the second and third case are left as an exercise.

Update: See the proof of the second case and third case.

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