Parallel ⇒ alternate interior angles equal

Dependencies: Parallel ⇒ corresponding angles equal (B.05, which grabs the corresponding angles for us), and Vertical angles are equal (to flip a corresponding angle into an alternate interior angle at $Q$).

Statement

Let $\ell_1 \parallel \ell_2$ be cut by a third line $t$: $t$ meets $\ell_1$ at $P$ and $\ell_2$ at $Q$. The two angles between $\ell_1$ and $\ell_2$ carved out by $t$, lying on opposite sides of $t$, form a pair of alternate interior angles. The two such pairs are respectively equal.

Let $\alpha$ denote a corresponding angle at $P$ (the angle formed on a specified side by $\ell_1$ and $t$), let $\beta$ be the corresponding angle at $Q$ on the same side as $\alpha$, and let $\gamma$ be the vertical angle to $\beta$ (i.e. the one at $Q$ that forms an alternate interior pair with $\alpha$). Then

Proof

The whole proof is just B.05 + vertical angles in two steps.

Step 1 (corresponding angles equal): by Parallel ⇒ corresponding angles equal, $\alpha = \beta$.

Step 2 (alternate and corresponding are vertical to each other): at $Q$, both $\beta$ and $\gamma$ have $Q$ as their vertex, and their sides are given by the two opposite rays from $Q$ along $\ell_2$ together with the two opposite rays from $Q$ along $t$ — this is exactly the definition of vertical angles. By Vertical angles are equal, $\gamma = \beta$.

Together, $\gamma = \beta = \alpha$. $\blacksquare$

Immediate consequences

• An alternate route to Parallel ⇒ co-interior angles supp. (B.07): from $\gamma = \alpha$, plus the fact that the co-interior and alternate interior angles at $Q$ form a linear pair (Linear pair sums to 180°), we immediately get $\alpha + \beta_{\text{co-int}} = 180^\circ$.
• The core lemma for the angle sum of a triangle = $180^\circ$: drawing through one vertex of the triangle a line parallel to the opposite side, this theorem is the unique tool that "transports" the three interior angles onto that parallel line.
• Construction ≡ algebraization of geometry: equal alternate interior angles let us copy any angle on $\ell_1$ to $\ell_2$ for free — together with the angle conditions of SSS / SAS, this is the entry point for many proofs of parallelogram properties, similar triangles, the Basic proportionality (intercept theorem) theorem (BPT), and so on.

Remarks

B.05 grabs the corresponding angles; this theorem flips them in place; pushing on to B.07 only needs one more linear pair. The three twin theorems (B.05 / B.06 / B.07) share the same "angle name converter" — that is why subsequent proofs cite them together so often.

Euclid's Elements I.29 lumps "equal alternate interior angles" and "equal corresponding angles" into a single proposition, and uses contradiction (assume not equal ⇒ co-interior angle sum $<180^\circ$ ⇒ the fifth postulate forces the lines to meet ⇒ contradiction). We split it into the three layers B.05 → B.06 → B.07; each layer is a direct construction, making the dependency chain unidirectional and contradiction-free.