Rule
An inscribed angle that spans the diameter of a circle is always ° .
This makes Thales’s theorem a special case of the inscribed angle theorem. If the inscribed angle is °, then the central angle is °. This is correct because the diameter can be thought of as an angle of °.
Think About This
From the figure, you can see that the triangles and are both isosceles triangles, because two of the sides of both triangles are the radius of the circle. You can therefore write the sum of the angles in the red triangle as follows:
Example 1
Find all the angles of the triangle, where is the diameter
Since is the diameter, Thales’s theorem tells us that . You also know that is the radius of the circle. Hence is isosceles and . You therefore find that
You know that is isosceles and that is the supplementary angle of . Hence
Furthermore,