Question

Prove that in any quadrilateral, the sum of the external angles at two vertices is equal to the sum of the internal angles at the other two vertices.

Answer 1

Consider quadrilateral PQRS:

In quadrilateral PQRS, by angle sum property, we have:

∠SPQ + ∠PQR + ∠QRS +∠RSP = 360°

⇒ ∠PQR + ∠QRS = 360° − ∠SPQ − ∠RSP ... (1)

Also, ∠QRS and ∠QRU are forming a linear pair.

⇒∠ QRS + ∠QRU = 180° ... (2)

Similarly, ∠PQT and ∠PQR are forming a linear pair.

⇒∠ PQT + ∠ PQR = 180° ... (3)

Adding equation (2) and equation (3):

⇒∠QRS + ∠QRU + ∠PQT + ∠PQR = 360°

⇒∠QRS + ∠PQR = 360° − ∠QRU − ∠PQT …(4)

From equation (1) and equation (4):

360° − ∠QRU −∠PQT = 360° − ∠SPQ − ∠RSP

⇒ ∠QRU + ∠PQT = ∠SPQ + ∠PSR

Hence, the sum of the external angles at the two vertices of a quadrilateral is equal to the sum of the internal angles at the other two vertices.