It All Came From Pencils

$\quad \quad \quad \quad AB=BD\quad and\quad BD=BC\quad \quad \\ \quad \quad \quad <BAD=<ADB=70°\\ and\quad <BCD=<BDC=70°\\ \quad \quad \quad \quad <ADF=CDG=90°\\ \therefore \quad x+90°+90°+70°+70°=360°\Rightarrow x=40°$

Imagine rays coming out of B passing through the center of each pencil. Obviously, they form an angle similar to $x$ as the rays are parallel to DF and DG. Moreover, each of the rays bisects corresponding tip of the pencil. Now we just add the inner bisected parts, which give us the answer: $20^{o}+20^{o}=40^{o}$ .

1. Draw lines from A to D and D to C
2. ABD and BCD will be isosceles triangles
3. angle BDA and angle BDC will be 70 degrees
4. angle GDC and angle FDA will be 90 degrees

SO.... 360 - (90 + 90 + 70 + 70) = 40 degree

Just for the fun of expanding the thinking...

Use 9 pencils (9 x 40° = 360°) and you will form a full circle made of its... and you will have the same angle x repeated 9 times. So if you form the same amount of angles than you have pencil the answer will be simply 40°.

I'm not great at maths so I don't understand what half of you are putting! I just did 40 + 40 divided by 2, I don't know if that's right, but that's what I did

Guys it's just your circle theorems. The the big angle, 40+40= 80 is then divided by 2 to get your answer of the smaller one. 80/2 = 40

Another similar pencil fits exactly in between these two pencils. This can be said by considering the bisectors of x. And, it turns out that they are equal to $20^o$ and $20^o$ , summing up to $40^o$ .

Simple - Symmetricity .

<BAD=<BDA=<BDC=<BCD (isosceles triangles) Hence,<BDA=<BDC=70°. Thus, <BDA + <BDC + <CDG + <ADF + <x = 360° => 70° + 70° + 90° + 90° + <x = 360° => <x = 360° - 320° .•. x = 40°

Imagine a pencil tip on it's side like a triangle... The angle of the tip is congruent to the angle of the sharpened space (alternate interior angles)... The thing is, the tip's at the center, so this angle is shared by both ends (20 degrees)... Combining the sharpened space of both pencils will be 40 degrees, same as the space they formed above...

There are 360/40 = 9 pencils that could fit like this in a circle. Between these 9 pencils are 9 x-degree openings. So, 9*x = 360 => x = 40 degrees.

Draw a line at the center of each pencil parallel to the body of the each pencil intersecting the tips of the pencils. Notice that the drawn lines are parallel to lines FD and Line DO. Now, also notice that the drawn lines are the halves of each of the following pencils. That gives the answer.

Well a straight line adds up to 180 degrees so think about the two pencils having a line between so their split so half of 80 is 40 as u got a split line that took the 100 degrees

A 2D pencil forms a pentagon. (Sum of Internal angles=540) So, <BDF=y 540=2y+40+180 y=160 Forming a circle centered in D. The sum of the internal angles is 360. 360=x+160+160 x=40

We can extend these two line backward than the angle formed by extended rays will be equal to x , than by taking BC as a arc x will be half of 80 degree.

between two pencils you can see the gap which resembles like big pencil tip and so it should also have same 40* :p .. helpful???

Easy the triangles ABD and DBC are equals and If you look at the triangle FDG is equal to the triangles ABD and DBC. Thus, If the triangles are equals the angles are equals, x = 40 degrees

Lets assume that E and F meet at infinity and G and H meet at infinity. Then <AED or < AFD =0. then < BAE = < BDF=160. (sum of angles in quadrilateral is 360 and by symmetry <BAE=<BDF).Similarly for other pencil. Then at point D, <BDF + < BDG + < FDG = 360. => x=40.

All angles of a pentagon sum up to 180*3.