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The theorem is named the Converse Consecutive Interior Angles Theorem because the same thing holds true but in opposite order.
The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of consecutive interior angles and one more angle.
It needs to be proven that $ℓ_{1}$ and $ℓ_{2}$ are parallel lines. It is already given that $∠1$ is supplementary to $∠3.$
Converse Corresponding Angles Theorem |
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. |
Since $∠1$ and $∠α$ are corresponding congruent angles, then $ℓ_{1}$ and $ℓ_{2}$ are parallel lines. This completes the proof.