Linear pair sums to 180°

Dependencies: Protractor axiom.

Statement

Let $O$ be a point on the line $\ell$, and let $C$, $D$ be points on $\ell$ on opposite sides of $O$; take a further point $E$ not on $\ell$. Then the two angles in the linear pair sum to

Here a "linear pair" means two angles sharing the vertex $O$ and one ray $OE$, while their other rays $OC$, $OD$ lie on the same line in opposite directions.

Proof

The proof uses only two direct consequences of Protractor axiom (axiom III).

Step 1: The angle between opposite rays is $180^\circ$.

Axiom III sets up a one-to-one correspondence between the rays from $O$ and the real numbers in $[0,\,2\pi)$; the angle between two rays equals the difference of the corresponding numbers (taking the representative in $[0,\,\pi]$).

For the two opposite rays $OC$, $OD$ from $O$ on line $\ell$, the difference of the numbers assigned by axiom III is exactly $\pi$. Hence

Step 2: Angle additivity splits $\angle COD$.

The ray $OE$ is not collinear with $\ell$ (since $E$ is not on $\ell$), so by the plane separation implied by the continuity of axiom III, the line $\ell$ divides the plane into two disjoint open half-planes, and $E$ must lie in one of them. The two cases are symmetric, so we may as well take $E$ to lie in the "upper" half-plane.

View $\angle COD$ as the $180^\circ$ straight angle "pointing into the upper half" — its interior is exactly the upper half-plane minus the point $O$. The ray $OE$ leaves $O$ pointing toward $E$, so it lies entirely inside this straight angle. By the angle additivity of axiom III,

Step 3: Combine.

Substituting step 1 into step 2:

Immediate consequences

• Entry point to triangle exterior angles: in $\triangle ABC$, extending the side $BC$ past $C$ to a point $X$, the angles $\angle ACB$ and $\angle ACX$ share the vertex $C$ and the ray $CA$, while their outer rays $CB$, $CX$ point in opposite directions along line $BC$. By the linear pair = 180° identity:

This is the entry point for Exterior angle equals the sum of two non-adjacent interior angles.

• Quantitative definition of "perpendicular": if at some point a pair of linear-pair angles formed by two lines are equal, then each angle equals $\dfrac{180^\circ}{2}=90^\circ$. This is precisely the standard definition of "$\ell\perp\ell'$".

• Entry point to the vertical-angle equality: applying linear pair = 180° to two pairs of linear-pair angles and subtracting immediately yields Vertical angles are equal — see the proof of the next theorem.

Remarks

Linear pair = 180° looks "obvious", but it actually formally welds together two facts:

• Axiom II tells us "two points determine a line" — but the line itself carries no information about "spanning $180^\circ$";
• Axiom III assigns each ray a number in $[0,\,2\pi)$ and stipulates that opposite rays correspond to numbers differing by $\pi$.

It is this convention of axiom III that makes "line = $180^\circ$" hold. Linear pair = 180° packages this convention together with angle additivity into a theorem that can be cited repeatedly; from this point on, every argument involving angle addition or subtraction (Vertical angles are equal, Triangle interior angles sum to 180°, Exterior angle equals the sum of two non-adjacent interior angles, Central angle is twice the inscribed angle on the same arc, …) starts from linear pair = 180°: cut a line at some point, and add or subtract the two resulting angles.