Question

Find the sum of all the angles at the five vertices of the adjoining star.

Answer 1

The given figure of a star is shown below.

To Find: ∠A + ∠B + ∠C + ∠D + ∠E

In triangle BEP, ∠B + ∠D = ∠QPD [Exterior angle property of triangles] … (1)

In triangle ACQ, ∠A + ∠C = ∠PQD [Exterior angle property of triangles] … (2)

Now, in triangle PQD, ∠QPD + ∠PQD + ∠D = 180° [Angle sum property of triangles]

⇒ (∠B + ∠E) + (∠A + ∠C) + ∠D = 180° [Using (1) and (2)]

∴ ∠A + ∠B + ∠C + ∠D + ∠E = 180°