Question

In the diagram, ABCDE is a regular pentagon. The sides AB and DC are produced to meet at F. Find the angles x, y and z.​

Answer 1

x = 108°

y = 72°

z = 36°

Explanation:

Finding unknown angles:

If all sides are equal in a pentagon, then it is a regular pentagon.

n = the number of sides = 5

Sum of all angles in a pentagon = (n -2) * 180

= (5 - 2)* 180

= 3 *180

= 540°

Sum of each angle of regular pentagon = 540 ÷ 5

= 108°

`\sf \boxed{\bf x= 108^\circ}`

Linear pair: If the non common arm of adjacent angles form a straight line, then the adjacent angles are called linear pair and they add up to 180°

∠DCB + y = 180 ° {Linear pair}

108 + y = 180

y = 180 - 108

`\sf \boxed{\bf y = 72^\circ}`

Exterior angle property of triangle: The exterior angle equals the sum of opposite interior angles.

In Δ BCF ,

x = y + z

Now, substitute the values of x and y , and we can find the value of z.

108° = 72 + z

108 - 72 = z

`\sf \boxed{\bf z = 36^\circ}`