What Does Thales's Theorem State?

Thales’s theorem on inscribed angle

Thales’s theorem on inscribed angle

Rule

Thales’s Theorem

An inscribed angle that spans the diameter of a circle is always 90° (u + v = 90°).

This makes Thales’s theorem a special case of the inscribed angle theorem. If the inscribed angle is 90°, then the central angle is 180°. This is correct because the diameter can be thought of as an angle of 180°.

Think About This

Proof of Thales’s Theorem

From the figure, you can see that the triangles BAP and CAP 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:

u + v + v + u = 180° 2u + 2v = 180° u + v = 90°

Q.E.D

Example 1

Example of Thales’s theorem

Find all the angles of the triangle, where AB is the diameter

Since AB is the diameter, Thales’s theorem tells us that ACB = 90°. You also know that SB = SC = SA is the radius of the circle. Hence BSC is isosceles and SCB = 35°. You therefore find that

BSC = 180° 35° 35° = 110°

You know that ASC is isosceles and that ASC is the supplementary angle of BSC. Hence

ASC = 180° 110° = 70°

Furthermore,

SAC = SCA = 180° 70° 2 = 55°

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