Exterior angle theorem

The exterior angle theorem can mean one of two things: Postulate 1.16 in Euclid's Elements which states that the exterior angle of a triangle is bigger than either of the remote interior angles, or 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: mACD = mABC + mBAC (here, mACD denotes the size of the angle ACD)

Proof:

Statement	Reason
In ∆ABC, ma + mb + mc = 180°------[1]	Sum of the measures of all the angles of a triangle is 180°
Also, mb + md = 180°-------[2]	Linear pair axiom
ma + mc + mb = mb + md	Transitive Property of Equality
ma + mc + mb = mb + md	Addition Property of Equality (also known as Subtraction Property of Equality)
∴ md = ma + mc
i.e. mACD = mABC + mBAC

Hence, proved.

References

• Geometry Textbook - Standard IX, Maharashtra State Board of Secondary and Higher Secondary Education, Pune - 411 005, India.
• Geometry Common Core, 'Pearson Education: Upper Saddle River, ©2010, pages 171-173 | United States.
• Wheater, Carolyn C. (2007), Homework Helpers: Geometry, Franklin Lakes, NJ: Career Press, pp. 88–90, ISBN 978-1-56414-936-7 .