Ratio of consecutive angles of a quadrilateral is 1:2:3:4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.
Suppose PQRS is a quadrilateral.
Let m∠P : m∠Q : m∠R : m∠S = 1 : 2 : 3 : 4
So, m∠P = k, m∠Q = 2k, m∠R = 3k and m∠S = 4k, where k is some constant
m∠P + m∠Q + m∠R + m∠S = 360º
∴ k + 2k + 3k + 4k = 360º
⇒ 10k = 360º
⇒ k = 36º
∴ m∠P = 36º
m∠Q = 2k = 2 × 36º = 72º
m∠R = 3k = 3 × 36º = 108º
m∠S = 4k = 4 × 36º = 144º
Now, m∠P + m∠S = 36º + 144º = 180º
We know if two lines are intersected by a transversal such that the sum of interior angles on the same of the transversal are supplementary, then the two lines are parallel.
∴ Side PQ || Side SR
Also, m∠P + m∠Q = 36º + 72º = 108º ≠ 180º
So, side PS is not parallel to side QR.
In quadrilateral PQRS, only one pair of opposite sides is parallel. Therefore, quadrilateral PQRS is a trapezium.