Inscribed Angle Theorem

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The Inscribed Angle Theorem is a theorem which states that any inscribed angle, that is, an angle with a vertex on a circle and whose sides are chords, measures exactly half the degree measure of the part of the circle inside the angle. This theorem is dependent on the measure of parts of circles in degrees, wherein the total circumference of any circle is 360 degrees.

Proof

This theorem is proven in three separate proofs:

• Proof that the measure of an inscribed angle, with one side a diameter, is half of the part of the circle inside the angle
• Proof that the measure of an inscribed angle containing the center is half the part of the circle inside the angle
• Proof that the measure of an inscribed angle excluding the center is half the part of the circle inside the angle

Proof the First

Assume a circle with inscribed angle ABC, segment BC a diameter. It is to be proven that the measure of ABC equals half the measure of the arc AC. This diameter must pass through the center, a point D, which splits BC into two equal parts. Connect D to A.

Because of how parts of circles are measured in degrees, the angle ADC is equal to the measure of the circle cut by ADC and by ABC, the arc AC.

By the Straight Angle Theorem, the angle ADC is equal to 180, minus BDA. By the Triangle Angle Sum Theorem, the sum of the angles ABD and BAD are equal to 180, minus BDA. Thus, by one of Euclid's axioms, the sum of the angles ABD and BAD is equal to ADC, as they are both equal to 180 minus BDA.

As both AD and BD are radii, they are equal, and thus by the Isosceles Triangle Theorem, the measures of angles ABD and BAD are equal to each other. As the sum of them is equal to ADC (which is the measure of the arc AC), each one is half of it, and as ABD and ABC are the same angle, the measure of angle ABC is half the measure of arc AC, Q.E.D.

Proof the Second

The proof of the second is more accurately a corollary of the first, in which a diameter is drawn within the inscribed angle, the first proof is applied as a lemma, and simple addition is performed.

Proof the Third

The proof of the third is more accurately a corollary of the first, in which a diameter is drawn within the inscribed angle, the first proof is applied as a lemma, and simple subtraction is performed.

References

• "9.5 Pt. 1: The Inscribed Angle Theorem." Beauty, Rigor, Surprise.