Construction of 145 degree angle

Question

I've tried doing it but I end up only constructing 135 degree angle.I have to use ruler without divisions and compass.It must be done with system of isosceles and equilateral triangle and their properties ,e.g external angle and etc. and the bisector.

Can you give me directions? Thank You in advance!

Answer 1

You cannot do it with an umarked ruler and compass.

Here is a way with a markable ruler and compass.

1. construct a 30 degree angle (bisect an equilateral triangle) and call the vertex $O$
2. draw a circle with centre $O$ and call where it meets the sides of the angle $A$ and $B$
3. mark the ruler with the radius of the circle
4. extend the line segment $AO$ beyond $O$, calling where it meets the circle again $C$, and then extend the line further beyond $C$
5. place the ruler so that it touches $B$, and so that it cuts the circle again and the extended line the distance apart marked on the ruler, then draw the line and calling these points $D$ and $E$ respectively
6. use what you know about angles and the isosceles triangles $BOD$ and $ODE$ to find the angle $OED$ (10 degrees)
7. add a right angle (90 degrees) and half a right angle (45 degrees) at E to get a total angle of 145 degrees ($10+90+45$)

Answer 2

If you can construct a 145 degree angle then you can construct a 55 degree angle by removing 90 degrees, and so a 10 degree angle, by removing 45 degrees. It is a classical theorem that a 10 degree angle cannot be constructed with ruler and compass. See http://en.wikipedia.org/wiki/Angle_trisection#Angles_may_not_in_general_be_trisected

Answer 3

At the level of elementary geometry, I'd guess that it was a typo and that you are expected to compute a 135 degree angle. It's not immediately obvious, but can be figured out pretty easily (as you have already done) once you figure out what 135 degrees really looks like.