Inscribed angle theorem

In geometry, a central angle is an angle whose vertex is at the center of a circle. A central angle is formed by two radii (plural of radius) of a circle. The central angle is equal to the measure of the intercepted arc. An intercepted arc is a portion of the circumference of a circle encased by two line segments meeting at the center of the circle

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Inscribed angle theorem

An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle. In the figure below, circle with center O has the inscribed angle ∡ABC. The other end points than the vertex, A and C define the intercepted arc AC of the circle.

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A

O

A

B

C

C

B

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y

D

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G

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x

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Theorem

The measure of an inscribed angle is half the measure of the intercepted arc.
Proof
Given Consider a circle C with center O , we consider an inscribed angle at B by the arc AC
To Prove
∡B= $\frac{1}{2}\measuredangle AOC$
Construction
Join the vertices A and C with center O. Also draw a line through B and O .

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A

O

A

B

C

C

B

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D

y

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b

SN	Statement	Reasons
1	∆BCO is Isosceles	CO=BO
2	y=2x	Triangle exteriar angle theorem
3	b=2a	Triangle exteriar angle theorem
4	y+b=2(a+x) a+x= $\frac{1}{2}(y+b)$ ∡B= $\frac{1}{2}\measuredangle AOC$	Adding 2 and 3

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Symbolic Notation

Due to the theorem given above, it is seen that, the measure of arc AC has equal influence to the measure of its central angle ∡AOC. So it is also written as
$\stackrel{⏜}{AC}\cong \measuredangle AOC$ or $\stackrel{⏜}{AC}\equiv \measuredangle AOC$
Similarly, the measure of chord AC has equal influence to the measure of its central angle ∡AOC. So it is also written as
$\overline{AC}\cong \measuredangle AOC$ or $\overline{AC}\equiv \measuredangle AOC$
Similarly, the measure of chord AC has equal influence to the measure of its arc AC. So it is also written as
$\overline{AC}\cong \stackrel{⏜}{AC}$ or $\overline{AC}\equiv \stackrel{⏜}{AC}$