I was given the following question:
PQRS is an isosceles trapezoid. If angle TRS= 100 degrees, angle T=35 degrees and PS=TS, find the measures of the numbered angles. Attached is a drawing of the diagram provided.
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I am having a lot of difficulty with this. I know that angle PRS will be 80 degrees because it is supplementary to angle TRS. But, I can’t figure out the rest. I see the relationships between the angles, etc, but I can’t figure out any of these measurements. I also don’t understand why the question provides the fact that PS=TS. That doesn’t seem to add anything useful. It can’t be telling me the triangles are congruent, because then angle PRS needs to be 100 degrees, which it is not.
Any thoughts?
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$\begingroup$As you noted, $\angle{PRS}=80°$. Take the triangle $\triangle{TRS}$: we know two angles out of three, so that $\angle{TSR}=45°$. Now take the isosceles triangle $\triangle{PTS}$: the two angles adjacent to the base $PT$ are equal to $35°$, so the third angle $\angle{PST}=110°$ and then $\angle{PSR}=65°$. This is one of the two angles adjacent to the base of an isosceles trapezoid: knowing it, you can easily complete the solution.
$\endgroup$ $\begingroup$The fact that $PS=TS$ provides you the information $\angle RPS=35^\circ$ and with this at hand, $\angle PSR=65^\circ$. Can you finish from here?
I have assumed that the points $P,R,T$ are collinear in accord with the figure.
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