## why is an inscribed angle in a semicircle a right angle?

Geometric formulae, word problems, theorems and proofs, etc.
king&i
Posts: 23
Joined: Sat Mar 07, 2009 5:13 pm

### why is an inscribed angle in a semicircle a right angle?

why is an inscribed angle in a semicircle a right angle? i know it is but im having trouble proving

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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Draw a circle, a diameter line, and an angle with its ends at the ends of the diameter line and its vertex on the circle. This is your inscribed angle, and it forms a triangle.

Draw a line from the vertex to the center of the circle. This is obviously a radius line, but it also splits the orignal triangle into two smaller triangles. On each of these triangles, two of their sides are radius lines, so the two triangles are isosceles.

Label the angles at the center as "A" and "B". Since a straight line (being the diameter line) has a "measure" of 180 degrees, then m(A + B) = 180 = m(A) + m(B). Whatever the other angles in the one smaller triangle measure, their measures are the same; label them as "C", so the triangle's angles are "A" and "C". Similarly, label the other angles in the other smaller triangle as "D".

We know that m(A) + 2m(C) = m(B) + 2m(D) = 180. Then:

. . . . .m(A) + 2m(C) + m(B) + 2m(D) = 360

By substitution, we get:

. . . . .(180) + 2m(C) + 2m(D) = 360

. . . . .2m(C) + 2m(D) = 180

Divide through by 2 to get:

. . . . .m(C) + m(D) = 90

What does this tell you about the measure of the original inscribed angle?

king&i
Posts: 23
Joined: Sat Mar 07, 2009 5:13 pm

### Re: why is an inscribed angle in a semicircle a right angle?

i never would of thought of doing the radius lines etc
thnx!