# What is measure of angle $x$

In the figure, $ABCDE$ is e regular pentagon and $FBC$ is a equaliteral triangle. What is measure of angle $x$

• Hint: $\triangle BFA$ is isosceles.
– dxiv
Commented Dec 9, 2016 at 17:06

Since two different answers are posted, I will chime in:

$$\angle ABC = 108^\circ$$

$$\angle FBC = 60^\circ$$

$$\implies \angle FBA = 48^\circ$$ $$\angle BFA + \angle BAF = 180^\circ- 48^\circ = 132^\circ$$

And since $\triangle FBA$ is isosceles, $$\angle BFA + \angle BAF =\frac{132^\circ}{2}= 66^\circ$$

• Thanky very much sir. Now i understand Commented Dec 9, 2016 at 17:32

Since we know that FBC is an equilateral triangle, we can deduce that FBA is an isosceles triangle.

This plus the facts, that every interior angle in a regular pentagon measures 108° and every interior angle in a equilateral triangle measures 60° gives us the final result:

x = 66°

• How find x=66, please told me, in detail form Commented Dec 9, 2016 at 17:22
• Which is the asnwers correct, x=66 or x=84 Commented Dec 9, 2016 at 17:29
• ∠CBF = 60° and ∠CBA = 108° so ∠FBA has to be 48°. Because the triangle FBA is an isosceles triangle we can deduce that BAF = \frac12 (180° - 48°) = 66° Commented Dec 9, 2016 at 17:34

IFF (If and only if) these conditions are met

All Sides are the same size therefore the equal angles to each angle in the regular rectangle are as follows: (excluding the equilateral triangle)

360/5 = 72*


Add the variable y and make it (Y is the angle that is the shortest in that triangle on the bottom right side of the regular pentagon)

72 - 60 = 12


now we know the new Variable Y, lets also conclude the second triangle is an iso-metric triangle. Meaning 2 of its sides are equal. Therefore:

180 - 12 = 2x


Since there are 2 same angles with the third angle known as 16* All we got to do now is solve for X:

180 - 12 = 168
168 = 2x

168/2 = 2x/2
x = 84


So X is 84 Degrees