Vertically Opposite Angles
Open a pair of scissors. Look at a railway crossing sign, or the two roads meeting at an X, or the
hands of a clock at twenty-past-eight. Every time two straight lines cross, they carve the space
around the crossing point into four angles — a big pair and a small pair, arranged in
an X.
Here's the neat part. The two angles sitting directly across from each other — sharing
only the crossing point, opening in opposite directions — are called vertically opposite
angles (or just "vertical angles"). And they are always exactly equal. Tilt the X
however you like; the angle in the top and the angle in the bottom stay a perfect match.
\text{vertically opposite angles are equal}
It follows from one fact you already know:
angles on a straight line
add up to 180^\circ.
When two straight lines intersect:
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the two pairs of opposite angles are equal —
a = c and b = d;
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each angle and the one next to it lie on a straight line, so
a + b = 180^\circ;
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so the four angles come in just two sizes, x and
180^\circ - x.
Why it works
No measuring needed — two straight lines do all the work. Step through the reason.
The trick is that a and c are both
"whatever is left after taking b away from
180^\circ" — so they have no choice but to be equal. The same argument works
for the other pair, giving b = d. So whenever lines cross, the angles
opposite each other match.
Worked examples
1) Find all four angles. Two lines cross, and one of the angles is
70^\circ. What are the other three?
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The angle vertically opposite it is equal: 70^\circ.
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Each angle next to the 70^\circ lies on a straight line with it,
so it is 180^\circ - 70^\circ = 110^\circ.
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And the fourth angle is vertically opposite one of those, so it is
110^\circ too.
Four angles, two sizes: 70^\circ, 110^\circ, 70^\circ, 110^\circ — and
they add to 360^\circ, a full turn, as they must.
2) Scissors and clock hands. When you open a pair of scissors so the blades make a
40^\circ angle, the two handles below the pivot make the
vertically opposite angle — also 40^\circ. Open the scissors wider and both
angles grow together, locked to the same value. It's the same on a clock: the tiny wedge between the
hands and the wedge on the exact opposite side of the centre are always equal.
3) A little algebra. Two lines cross. One angle is written
(2x)^\circ and the angle vertically opposite it is
(x + 35)^\circ. Find x.
Vertically opposite angles are equal, so set them equal:
2x = x + 35 \;\Rightarrow\; x = 35.
Each angle is therefore 2 \times 35 = 70^\circ — and a quick check,
x + 35 = 70^\circ, agrees. Equal angles let you turn a picture into an
equation.
The name fools almost everyone. "Vertically opposite" does not mean up-and-down or
pointing to the sky. It comes from vertex — the crossing point — and means "opposite you
through the crossing point." The two equal angles can both be lying nearly flat, or tilted at
any jaunty angle; what matters is that they face each other across the vertex.
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Equal = across, not next to. The pair that are equal are the ones not
touching each other. The angle right beside a given angle is a different relationship: they
sit on a straight line and are supplementary — they add to
180^\circ, they are not equal (unless both happen to be
90^\circ).
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Don't grab the neighbour. If one angle is 50^\circ, its
vertical opposite is 50^\circ — not 130^\circ.
The 130^\circ is the one next to it.
This is one of the very first things anyone proved in all of geometry. Around 300 BC — over two
thousand years ago — Euclid wrote it down as
Proposition 15 of Book I of his Elements, the most successful textbook in
history. He didn't just measure a few crossings and shrug; he proved that opposite angles
must be equal, for every pair of lines that ever could be drawn, using exactly the straight-line
argument you just stepped through.
It's a small fact, but it's a foundation stone: once you trust it, you can start
angle-chasing — the detective game of unlocking every angle in a complicated figure
from just a few given ones — the whole art of
angle chasing.
Practise: chase the angles
Two lines cross. Fill in every angle you can — using vertically opposite angles
and angles on a straight line — ending with the highlighted one.
Refresh for a new figure; Check explains each step.
See it explained