Parallel ⇒ co-interior angles supplementary

Dependencies: Parallel ⇒ corresponding angles equal (Parallel ⇒ corresponding angles equal) and Linear pair sums to 180° (Linear pair sums to 180°).

Statement

Let two lines $\ell_1\parallel\ell_2$ be cut by a third line $t$, meeting at $P\in\ell_1$ and $Q\in\ell_2$. The two interior angles enclosed by $\ell_1$ and $\ell_2$ on the same side of $t$ are called a pair of co-interior angles (also "consecutive interior angles"). Conclusion: each pair of co-interior angles is supplementary, i.e.

Proof

The strategy is to stitch "corresponding" and "co-interior" together at $Q$: the corresponding-angle equality comes from Parallel ⇒ corresponding angles equal, the linear-pair equality from Linear pair sums to 180°, and adding the two yields the result.

Step 1 (corresponding angles): by Parallel ⇒ corresponding angles equal, when $\ell_1\parallel\ell_2$ is cut by $t$, corresponding angles are equal. Let $\alpha$ be some interior angle at $P$; its corresponding angle at $Q$ is $\alpha'=\alpha$.

Step 2 (linear pair): at $Q$, $\alpha'$ and the co-interior angle $\beta$ (on the same side of $t$) share the vertex $Q$ and a common side along $t$; their other sides are opposite rays from $Q$ along $\ell_2$ — the hypothesis of Linear pair sums to 180° is satisfied. Hence

Step 3 (substitute): substituting $\alpha'=\alpha$:

Immediate consequences

• Same-side exterior angles supplementary: the same-side exterior angles are the vertical angles of the $\alpha$, $\beta$ above (by Vertical angles are equal), so the same-side exterior angle sum is also $180^\circ$.

• Criterion ⇔ property — the full set is in: together with Parallel ⇒ corresponding angles equal (corresponding angles) and Parallel ⇒ alternate angles equal (alternate interior angles), the "three pairs of angles" criteria and properties for parallel lines are now symmetrically complete: parallel ⇔ corresponding angles equal ⇔ alternate interior angles equal ⇔ co-interior angles supplementary.

• A rectangle has $90^\circ$ adjacent angles: in a rectangle the two pairs of opposite sides are parallel; adjacent angles are co-interior, so the adjacent-angle sum is $180^\circ$; combined with Parallelogram: opposite angles equal (Parallelogram: opposite angles equal), all four angles are $90^\circ$. This is the key step in the "$(c)\Rightarrow$ rectangle" branch of rectangle-equiv-conditions.

Remarks

The "main line" of this property — corresponding-angle equality + linear-pair equality + substitution — is in fact a template shared by every property of parallel lines: first build a "transport bridge" for angles between $\ell_1$ and $\ell_2$ (corresponding / alternate), then locally at $Q$ use Linear pair sums to 180° to combine the angles into the desired form. Using alternate interior angles as the bridge (Parallel ⇒ alternate angles equal) gives the same conclusion; we choose Parallel ⇒ corresponding angles equal because it sits closest to the axioms in the dependency graph, giving a shorter chain.