Geometry Problem: ∠EDB in Triangle ABC

[projjal]

Homework Statement

Suppose ABC is a triangle, where ∠ABC = ∠ACB =80°. A line BD through B meets AC at D such that
∠DBC=60°.Similarly CE meets AB at E where ∠ECB = 50°. Whats the value of ∠EDB?

Homework Equations

The Attempt at a Solution

I solved the problem using coordinate geometry but i was asked to solve it using simple geometry.

 

[tiny-tim]

hi projjal!

… i was asked to solve it using simple geometry.

show us your diagram, with all angles marked, and all equal distances marked

 

[projjal]

sorry for that. i have uploaded the diagram.

 

[micromass]

Let's denote the intersection of EC and BD by F. Can you find the angles at this point F?

Can you find the angle at A?

 

[tiny-tim]

i have uploaded the diagram.

mark all the angles, and all equal distances

(and preferably re-draw it so that those 80° angles look like 80°)

 

[projjal]

preferably re-draw it so that those 80° angles look like 80°

I think now it should be okay with the diagrams.I have also marked all the angles.
here also AB=AC & AD=BD
now what's EDB ?

 

[tiny-tim]

ah , that's much better!

here also AB=AC & AD=BD

also BE = BC

hmm … i can't see any obvious way to find EDB ​

(longshot: i wonder whether it has anything to do with 40° being twice 20° ?)

 

[micromass]

Draw a line in D parallel to BC.

 

[pbuk]

If you extend DE and CB until they meet at a point P, can you find angle EPB?

 

[projjal]

If you extend DE and CB until they meet at a point P, can you find angle EPB? .

I have already tried that but i couldnot find the answer.

 

[projjal]

Draw a line in D parallel to BC.

I couldnot get anything from that.If you could do it please specify it to me.

 

[pbuk]

EPDF form a quadrilateral, the other 3 angles are known.

 

[projjal]

EPDF form a quadrilateral, the other 3 angles are known.

If P is the point of intersection of DE & CB how EPDF forms a quadrilateral?

 

[pbuk]

If P is the point of intersection of DE & CB how EPDF forms a quadrilateral?

Sorry, EPBF - not enough screen area to see the diagram and type at the same time!

 

[haruspex]

Sorry, EPBF - not enough screen area to see the diagram and type at the same time!

We know PBF and BFE. What's the third?
I don't think it's going to be that simple anyway. Seems to me these angles have been chosen quite carefully. It's no coincidence that there are isosceles triangles all over the place which would not be so if we were to change just one of the given angles.

 

[phydis]

this might help you-> http://www.duckware.com/tech/worldshardesteasygeometryproblem.html
but that is not the only way to solve this.

 

[projjal]

So that's how its done.

 

[pbuk]

We know PBF and BFE. What's the third?

Oops, clearly not a good idea for me to try and do geometry in my head. I see the solution is not trivial.