Question

The angles of a polygon are x , (x – 24) , (x + 20) , (2x – 38) and (2x – 69) . Find x and hence state the measure of the greatest angle​

Answer 1

Concept used

As we know that we are given with the hints of the measurements of all the angles of a polygon. But, we don't know that which polygon it is. As we are given with five angles of the polygon, we can easily state that the given polygon to us is the pentagon which has five angles and five sides. To find the answer of this question, we use the angle sum property of the pentagon. It helps us to find the measurements of all the angles of this polygon.

Measurement of first angle :

$\sf \dashrightarrow {Angle \: sum \: property}_{(Pentagon)} = {540}^{\circ}$

$\sf \dashrightarrow x + (x - 24) + (x + 20) + (2x - 38) + (2x - 69) = {540}^{\circ}$

$\sf \dashrightarrow 7x + (-24) + 20 + (-38) + (-69) = {540}^{\circ}$

$\sf \dashrightarrow 7x - 24 + 20 - 38 - 69 = {540}^{\circ}$

$\sf \dashrightarrow 7x - 4 - 107 = {540}^{\circ}$

$\sf \dashrightarrow 7x - 111 = {540}^{\circ}$

$\sf \dashrightarrow 7x = 540 + 111$

$\sf \dashrightarrow 7x = 651$

$\sf \dashrightarrow x = \dfrac{651}{7}$

$\sf \dashrightarrow x = 93$

Now, we can find the measurement of all the angles.

Measurement of second angle :

$\sf \dashrightarrow x - 24$

$\sf \dashrightarrow 93 - 24$

$\sf \dashrightarrow {69}^{\circ}$

Measurement of third angle :

$\sf \dashrightarrow x + 20$

$\sf \dashrightarrow 93 + 20$

$\sf \dashrightarrow {113}^{\circ}$

Measurement of fourth angle :

$\sf \dashrightarrow 2x - 38$

$\sf \dashrightarrow 2(93) - 38$

$\sf \dashrightarrow 186 - 38$

$\sf \dashrightarrow {148}^{\circ}$

Measurement of fifth angle :

$\sf \dashrightarrow 2x - 69$

$\sf \dashrightarrow 2(93) - 69$

$\sf \dashrightarrow 186 - 69$

$\sf \dashrightarrow {117}^{\circ}$

As we can see that the fourth angle is greatest among all of them.

Hence, the fourth angle measuring 148° is the greatest angle and the value of x is 93.