I found this problem in my geometry textbook and Iâ€™ve been staring at the picture for a while and I am out of ideas.

According to the official answers sheet the answer is 15 degrees given in sexagesimal units.

The rules for this is that you may not use trigonometry or triangle similarity. So in essence is a puzzle. Well, i am not good with constructions. Does any of you guys can help me?

You can use isosceles triangles, circles, equilateral triangles, congruence and all of those sorts. Well, not the secant tangent formula, but I doubt that it will be needed.

Since this problem is in a section where it only introduces concepts like midpoint of a triangle, congruence cases, and basic theorems like incenter, circumcenter, barycenter and orthocenter. Maybe using these simple tools it can be solved.

Therefore, Is there a math wiz who can someone help me?. Please say it with some logical steps without skipping something so I donâ€™t get lost.

Thereâ€™s two unknown angles in there. Assign them variable names, then you can form three simultaneous equations in three variables using the basic properties of a triangle and square.

edit: What I wrote was nonsense. As shown by the pictures Ender drew: How do you find the angle inside a kite? - #71 by Ender. It probably only works for x = 15 degree. Hopefully someone educated can explain how to reach the solution. Considering the trickiness of the problem, itâ€™s probably contest math question.

But you havenâ€™t got â€śthereâ€ť. It should be 2x + y instead of 4x + y, where y is the angle ABD. And angle DBC should be 3x.

Anyways, it doesnâ€™t tell us much. It just gives us the equation y + 6x = 180. Which we could already tell from triangle ABD.

I suspect the solution is that line equation y + 6x = 180 subject to certain constraints like, y and x should be positive, and the interior angles should be between 0 and 180 degrees.

Because if we follow the suggested solution, x is 15 degrees, and y would be 90 degrees. But other pairs like x = 10 deg and y = 120 deg, etc. seem to work just fine.

Perhaps the question is missing certain information. But then again, I have no idea what â€śdegrees given in sexagesimal unitsâ€ť means.

Well it looks like angle A has to be 3x, and so is angle D, and so it follows that A and D is the same value. angle C has to be 2x and therefore assuming angle C is 30 degrees (which is what it would be if the answer is 15), then angle A and D must both be 45 degrees. But then that means the remaining angle of triangle ABD must be 90 degrees but the text does not indicate that it is (if you have a triangle with 2 45 degree angles then the last angle MUST be 90 degrees, otherwise it wonâ€™t work), but inferring from the diagram it looks like angle B must be an acute angle if you donâ€™t include the other side of the triangle.

In the end you should probably plug this into wolfram alpha and see what it tells you, or use calculus.

This is confusing. I thought that the way we normally express degrees as degrees, minutes, seconds already was expressed as sexagesimal units, as opposed to being expressed as, say, radians.