 # Circle Theorem Examples

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In this lesson, we will learn

• a Circle Theorem called the Thales' Theorem or Triangle inscribed in semicircle or Angle inscribed in semicircle or “90 degrees in Semicircle” Theorem.
• how to use the Thales' Theorem to find missing angles.
• how to prove the Thales' Theorem.

### Thales' theorem

The Thales' theorem states

Every angle subtended at the circumference by the diameter of a circle is a right angle (90˚).

or

The diameter of a circle always subtends a right angle to any point on the circle.

or

The angle inscribed in a semicircle is 90˚.

The following diagram shows the Thales' Theorem: Angles in a semi-circle are 90°. Scroll down the page for more examples and solutions. ### Using the Theorem

Example:

O is the centre of the circle. Find the value of x Solution:

ABC = 90˚ ( angle in a semicircle = 90˚)
63˚ + 90˚ + x = 180˚ ( sum of angles in a triangle )
x = 27˚

Inscribed Right Triangles (Right Triangles Inside of Circles)
Thales' Theorem: If the longest side of a triangle inscribed within a circle is the same length as the diameter of a circle, then that triangle is a right triangle, as well as the converse: if a right triangle is inscribed within a circle, the length of its hypotenuse is the diameter of the circle.
We can use the Thales' Theorem to find missing angles.

### Proving the Theorem

Proof of the Thales' Theorem Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle.
This proof uses the bow theorem.

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