Some things to consider:
-This theorem has proved very, very many theorems many of which with trig, so you can't use any theorems that have been proven with the triangle sum theorem.
-The Triangle Sum Theorem states that the measures of the angles in a triangle (in degrees) must add to $180^\circ$.
This is a theorem therefore it already has been proven, so what would that proof be? I duly noted that this is an elementary topic, but it seems that all my ways of proving it failed because at lease one of the steps in the proof had a reason that had been proved already by none other than the $\Delta $ sum theorem.
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$\begingroup$I will assume you are working in Euclidean Geometry (since this is untrue in Non-Euclidean Geometry).
In particular, you have the use of the fifth postulate which reads:
"If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles."
You could work in a setting where this is untrue (as detailed in the wiki-link above).
Let $\overline{AB}\parallel \overline{CD}$. You should know then that $\angle ABC = \angle DCB$. You should also know that $\angle CAB = \angle ECD$.
By the definition of a right angle, $\angle ACE = 180^\circ = \angle BCA + \angle DCB + \angle ECD = \angle ACB + \angle CBA + \angle BAC$
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