Geometric Proofs Math Example 2

Solution

1. 1
   Assume lines $\ell$ and $m$ are cut by transversal $t$ at points $P$ and $Q$, with alternate interior angles $\angle 1 = \angle 2$.
2. 2
   Suppose for contradiction that $\ell$ and $m$ are not parallel. Then they meet at some point $R$, forming triangle $PQR$.
3. 3
   In triangle $PQR$, the exterior angle at $P$ (which equals $\angle 1$) must be greater than the non-adjacent interior angle at $Q$ (which equals $\angle 2$) by the Exterior Angle Theorem.
4. 4
   But this contradicts $\angle 1 = \angle 2$. Therefore the assumption is false, and $\ell \parallel m$.