# Help with a logic question

$\angle B_1 + \angle B_2 = 180^\circ$

$\angle C + \angle B_1 = 180^\circ$

$\angle D + \angle B_2 = 180^\circ$

Can I prove with these 3 statements that:

$\angle D = \angle C$?

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## 2 Answers

It is not possible to prove $\angle D = \angle C$. Here is a counterexample:

\begin{align} \angle B_1 & = 45^\circ \\ \angle B_2 & = 135^\circ \\ \angle C & = 135^\circ \\ \angle D & = 45^\circ \end{align}

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Rearranging each of the first two equations gives $\angle B_2 = 180^\circ - \angle B_1$ and $\angle C = 180^\circ - \angle B_1$ respectively, and so $\angle B_2 = \angle C$. Putting this into the third equation, we have $\angle D + \angle C = 180^\circ$. This does not imply that $\angle D = \angle C$, unless both are equal to $90^\circ$.

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