Definitions

exterior angle

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 + mBAC (here, mACD denotes the size of the angle ACD)

Proof:

Statements Reason
In ∆ABC, m'a + m'b + m'c = 180°------[1]''' Sum of the measures of all the angles of a triangle is 180°
Also, m'b + m'd = 180°-------[2] Linear pair axiom
m'a + m'c + m'b = m'b + md From [1] and [2]
m'a + m'c + m'b = m'b + md
m'd = m'a + mc
i.e. m'ACD = m'ABC + mBAC

Hence, proved.

References

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