The easy rule that comes to mind is all 3 angles of a triangle add up to 180 degrees. I believe you can make simultaneous equations from observations of several triangles.
* Total angle B = 180-3x
* Triangle ABD is isosceles (because x+2x=3x). Line AB=BD.
* Unknown angle ABD=180-6x
* Triangle ACD is isosceles. Line AD=DC.
* Unknown angle BDC =180-5x.
* Line CA goes through midpoint of BD (because x+x=total angle C).
I'm not certain but no doubt a few more observations and some deductive reasoning ought to lead to a provable solution.
Yes, I recall the identity you mention. For any polygon the sum of their interior angles is given by,
S=180(n-2), where n=sides
But this problem is a little bit more complicated. It requires a construction. Otherwise it cannot be solved. If somebody has ideas on how or what to do, it would be of great help.
By the way, I disagree. AC does not passes by the midpoint of BD. Although the angle bisector theorem in a triangle states something like this is bounded by the fact that it must be
isosceles and triangle BCD is not isosceles or at least we don't have sufficient information to affirm this.
Still anyone can help here?. I am stuck. How would you solve this without requiring similarity and only congruence or simple constructions?.
@BradtheRad - that is exactly how I started to answer this question but then I saw that we are not to use trig. Therefore I'm not sure that this is the approach the OP is looking for.
On the other hand, why make the problem harder than it needs to be!!! Unless this is a homework problem to test the understanding of something taught in class.
Susan
Thank you for your valuable feedback, it is just that this problem belongs to my geometry textbook and I am not sure how to solve it. Surely is challenging, but that's how it is. Sorry about that.
Other than this, what do you suggest?. By the way, when I meant that trigonometry functions are not allowed, it is because this problem is intended to be solved without them.
The intended approach is relying in synthetic geometry or euclidean geometry as it is known more commonly, and that is relying in euclidean constructions like lines, circles and all that stuff. Thus, as I mentioned above, does any of you guys have any idea on what to do?.