Block, SkipCaptcha, Automoderated users, edit
1,942
edits
Changes
Jump to navigation
Jump to search
A page about a theorem in geometry
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 [[degree|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''.
[[Category:Geometry]]
==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''.
[[Category:Geometry]]