### Exterior Angle Property of a Triangle: Proof Of Exterior Angle Theorem

In this module, we will learn about �Exterior Angle Property of a Triangle�.

Draw a triangle ABC and produce one of its sides, say BC. Observe the angle ACD formed at the point C. This angle lies in the exterior of triangle ABC. We call it an exterior angle of the triangle ABC formed at vertex C. Clearly angle BCA is an adjacent angle to angle ACD. The remaining two angles of the triangle namely angle A and angle B are called the two interior opposite angles or the two remote interior angles of angle ACD.

Now cut out (or make trace copies of) angle A and angle B and place them adjacent to each other as shown in figure. Do these two pieces together entirely cover angle ACD? Can we say that measure of angle ACD is equal to the sum of measures of angle A and B?

Yes, because these two angles entirely cover angle ACD.Thus, an exterior angle of a triangle is equal to the sum of its interior opposite angles.

## Exterior Angle Property of a TriangleProof

Let us justify this mathematically in the form of a statement.

Statement: An exterior angle of a triangle is equal to the sum of its interior opposite angles.

It is given that a triangle ABC, in which angle ACD is an exterior angle.

We need to prove that angle ACD is equal to the sum of angles A and B.

Let us perform construction for this proof.

Through vertex C draw CE, parallel to BA.

Let us start with the justification of the Exterior Angle Property of a Triangle.

In figure angle 1 is equal to angle x, this is equation (i)

Because BA is parallel to CE and AC is a transversal.

Therefore, alternate angles should be equal.

And angle 2 is equal to angle y, this is equation (ii)

As BA is parallel to CE and BD is a transversal.

Therefore, corresponding angles should be equal.

Adding equations (i) and (ii), we get , ?1 + ?2 = ?ACD

Thus we can see that sum of two opposite interior angles is indeed equal to the exterior angle. The above relation between an exterior angle and its two interior opposite angles is referred to as the Exterior Angle Property of a triangle.