In the figure below, how do I find the value of the variable?

ns.19

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When I try to solve using the formula for the sum of the internal angles successively, I find a linear system with infinite solutions. Where am I wrong?
 

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Using the fact that the BDA triangle is isosceles and using the external angle theorem, can I conclude that the ADC angle measure is 80?
 
If you show us your work then we can see if and where you are wrong? Please post your work as the forum guidelines request.

For the record I too get an infinite solution. I of course looked at BDA being isosceles but that did not help.
I recall a similar problem like this where Dr Peterson showed that there is a unique solution. If there is one I am sure that someone will point this out and state why. Meanwhile keep trying.
 
Right. First do: measure the angle ADC = d, measure the angle BDC = z and measure the angle BCD = y. Then, using the sum of the internal angles theorem in triangle ABC, we conclude that x + y = 50, using the same theorem in triangle BDC, we conclude that z + y = 110, using the same theorem in triangle ADC, d + x = 140. Finally, looking at the figure we have 160 + d + z = 360, that is d + z = 200. Setting up the linear system with this information, we can see that such a system has infinite solutions.
 
Right. First do: measure the angle ADC = d, measure the angle BDC = z and measure the angle BCD = y. Then, using the sum of the internal angles theorem in triangle ABC, we conclude that x + y = 50, using the same theorem in triangle BDC, we conclude that z + y = 110, using the same theorem in triangle ADC, d + x = 140. Finally, looking at the figure we have 160 + d + z = 360, that is d + z = 200. Setting up the linear system with this information, we can see that such a system has infinite solutions.
Yes, that is just what I got, just with different lettering. I agree that the system you came up with does not have a solution. However what you have is more than just a system of equations, you have a geometric figure that can influence things. For example, d, z and y can't be negative. Possibly there is something else which we are missing that can make this problem uniquely solvable.
 
I understand, but, this question was passed on to me only in this way. Without any more information.
 
Yes, that is just what I got, just with different lettering. I agree that the system you came up with does not have a solution. However what you have is more than just a system of equations, you have a geometric figure that can influence things. For example, d, z and y can't be negative. Possibly there is something else which we are missing that can make this problem uniquely solvable.
I think there is a unique solution (by construction?): ABC has angles 50, 50, 80. Point D is uniquely defined by the 10 degree angles. Therefore angle x is uniquely defined.
If you drop heights from D to AC and AB and make |AB| = 1 you can calculate everything using sin and cos of known angles. There is probably an easier way to do it.
 
I understand, but, this question was passed on to me only in this way. Without any more information.
I am not saying that you did not supply all the information. I am saying that possibly we are missing something that is a result of what you gave us.
 
I think there is a unique solution (by construction?): ABC has angles 50, 50, 80. Point D is uniquely defined by the 10 degree angles. Therefore angle x is uniquely defined.
If you drop heights from D to AC and AB and make |AB| = 1 you can calculate everything using sin and cos of known angles. There is probably an easier way to do it.
Nicely done! I am embarrassed at how poorly I do with geometry. I really have no talent in this subject. Your method was so obvious. Thanks for pointing it out to me and the OP and increasing my geometry maturity!
 
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