Lesson Angle in a semicircle

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Theorem: An angle inscribed in a Semi-circle is a right angle.

In the above diagram, We have a circle with center 'C' and radius AC=BC=CD. Angle inscribed in
semi-circle is angle BAD.

To proof this theorem,

Required construction is shown in the diagram. Draw the lines AB, AD and AC.
Now there are three triangles ABC, ACD and ABD.
Angles of the triangles are shown in the diagram.

Observe that triangle ABC and ACD are Isosceles triangle.
In triangle ABC
BC=AC
Angles in front of these lines are also same (Call it x)............(Property of isosceles triangle)

Similarly,
In triangle ACD
AC=DC
Angles in front of these lines are also same (Call it y)............(Property of isosceles triangle)

Now again consider triangle ABC:
Sum of all three angles in the triangle is 180.............................(Property of triangle)
Hence x%2B+y+%2B+w=+180
w+=+180-%28x%2B+y%29..........................................................(1)

Similarly,
In triangle ACD:
x%2B+y+%2B+z=+180
z+=+180-%28x%2B+y%29..........................................................(2)

After adding equation number (1) and (2),

w+%2B+z=+360+-+2%2A%28x%2By%29..................................................(3)

Observe, BCD is a straight line. Hence, Angle(BCD) is a straight angle.

w+%2B+z+=+180.............................................................................................(Property of Straight Angle)

Now, plug the valve of Angle(w+z) in equation number (3)

180+=+360+-+2%2A%28x%2By%29
2%2A%28x%2By%29=180
x%2By+=90
Hence, Angle(BAD) is right angle.
Proof Ends!!!

For more information on Circle refer to wikipedia.
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