# Finding angles of triangles

I have what appears to be a $3$-sided triangle: it is two lines on a 180 degree line at the bottom. The bottom left angle is $4x-3$ the top angle is $6x + 3$ and the bottom right angle is not given, but on the angle outside of the triangle between the outside of the triangle and the line is $9x+12$. What am I suppose to do?

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Set up a system of equations. You know that the angles of the triangle add up to $180$, and that the angle of a line is also 180. Hence the line at the bottom is equal to $$\theta = 180 - (9x+12).$$

Now you have $$(4x - 3) + (6x +3) + \theta = 180.$$

Solve for $x$, substitute back into each angle measure, and you are finished.

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Not getting any of this maybe I can try and draw a triangle. ^ there, that works like the bottom is a straight line and the last angle is outside of the triangle. – Adam Jun 1 '11 at 2:24
@JTL: Punctutation should go inside displayed formulas, $$xxxx.$$, not outside, to prevent the orphaned periods. The general rule is: punctuation goes outside in-line formulas, inside displayed formulas. – Arturo Magidin Jun 1 '11 at 3:07
Let $\theta =$ (bottom right angle of triangle). $9x + 12$ is the measure of the angle "supplementary" to $\theta$ (angle on outside the triangle formed by the right side and the line extended to the right from the bottom side of the triangle. Since two adjacent (next to each other) angles form a line (180 degrees), we get that $\theta + (9x + 12) = 180. So$\theta = 180 - (9x + 12). Then in the second equation above, "plug in" $180 - (9x+12)$ for $\theta$. – amWhy Jun 1 '11 at 3:09
JTL: if you look at the other answer, in one of the comments from the OP, the value of the top angle has changed (otherwise I was getting x = -45). With top angle at 6x + 3, the solution is x = 12. Just in case you want to update your lay-out of the problem – amWhy Jun 1 '11 at 3:37
@amWhy: I think I've corrected it as it ought to be? Sorry, OP is a bit confusing; regardless, I think he figured it out. @Arturo: Thanks, duly noted. – barf Jun 1 '11 at 4:41

As opposed to those 4-sided triangles? :-)

From your description of the triangle, and the fact that the sum of the interior angles of a triangle is always equal to $180$, we have the equation

$$(4x-3)+(2x-120)+\theta=180,$$

where $\theta$ is the angle we don't know the value of yet.

But, we do know that the supplement of the angle $\theta$ is $9x+12$. Therefore $$\theta=180-(9x+12)=-9x+168.$$ Now plug this into the previous equation and solve for $x$.

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As opposed t o a 3 sides shape that is not a triangle. – Adam Jun 1 '11 at 2:20
Could you draw me a picture of a 3-sided polygon that isn't a triangle? =) – barf Jun 1 '11 at 2:21
I don't get it, how is the supplement that? – Adam Jun 1 '11 at 2:21
3 sided shape that isn't a triangle would be a triangle where a= 5 b=3 and c=8 – Adam Jun 1 '11 at 2:22
Adam, you need to correct your values entered for the angles in the question: you just gave a different value for the top angle in your question (2x -120) but just said it's 6x + 3, which makes more sense! – amWhy Jun 1 '11 at 3:24

I'm including a picture to help "spell things out" visually and for a visual example of "supplementary angles." I've worked the problem out on the image as well.

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Thanks, Amzoti! – amWhy May 23 '13 at 1:51