Exterior angle theorem

Exterior angle theorem

The exterior angle theorem is a theorem in elementary geometry which states that the exterior angle of a triangle is equal to the sum of the two remote interior angles.

A triangle has three corners, called "vertices". The sides of a triangle that come together at a vertex form an angle. This angle is called the "interior angle". In the picture below, the angles "a", "b" and "c" are the three interior angles of the triangle. An "exterior angle" is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle "d" is an exterior angle.

The exterior angle theorem says that the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle. So, in the picture, the size of angle "d" equals the size of angle "a" plus the size of angle "c".



Proof

Given: In ∆"ABC", angle "ACD" is the exterior angle.

To prove: "m'ACD" = "m'ABC" + "m"BAC (here, "m""ACD" denotes the size of the angle "ACD")

Proof:

Hence, proved.

References

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