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

I didnt now how to find. Please help me. Thanku very much enter image description here

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

3 Answers 3

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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, $$\angle BFA + \angle BAF =\frac{132^\circ}{2}= 66^\circ$$

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  • $\begingroup$ Thanky very much sir. Now i understand $\endgroup$
    – brain host
    Commented Dec 9, 2016 at 17:32
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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°
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  • $\begingroup$ How find x=66, please told me, in detail form $\endgroup$
    – brain host
    Commented Dec 9, 2016 at 17:22
  • $\begingroup$ Which is the asnwers correct, x=66 or x=84 $\endgroup$
    – brain host
    Commented Dec 9, 2016 at 17:29
  • $\begingroup$ ∠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° $\endgroup$
    – jonask
    Commented Dec 9, 2016 at 17:34
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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

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