Question

The side EF, FD, and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ, and FER respectively. Prove that...

Answer 1

The sum of the exterior angles of a triangle, when the sides are produced in order, is twice the sum of the interior angles, which in value, equates to 360 degrees. This is because any exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it.

Step-by-step explanation:

In the triangle DEF, let's consider the exterior angles DFP, EDQ, and FER. In geometry, we know that the sum of interior angles of a triangle is 180 degrees. Considering Triangle DEF, the interior angles are ∠EFD, ∠FDE and ∠DEF. One of the key properties of triangles is that any exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it.

So:
Angle DFP = Angle FDE + Angle DEF
Angle EDQ = Angle DEF + Angle EFD
Angle FER = Angle EFD + Angle FDE.

And when we add all three exterior angles together: Angle DFP + Angle EDQ + Angle FER = 2(Angle DEF + Angle EFD + Angle FDE)
It means the sum of the exterior angles, when the sides are produced in order, is twice the sum of the interior angles, which is 2 * 180 = 360 degrees. Hence proved.

Learn more about Exterior Angles of a Triangle