Two lines crossing at O. The opposite pair ∠1 = ∠3 (one tick) and the opposite pair
∠2 = ∠4 (two ticks); and any neighbor pair, like ∠1 + ∠2, makes a straight line: 180°.
In Stage 13 you measured a single angle, one ray swinging away from another. Now watch what happens
when two straight lines meet. Four angles appear at the same instant — but you do not get to choose them freely.
Two plain-as-day rules do all the work: neighbors on a line fill a straight angle, and opposite angles
match. Together they lock the whole family so tightly that knowing one of the four angles quietly hands you
the other three. By the end of this lesson you will read "because" into a figure for the first time.
14.1.1 Two lines, one crossing point
Two distinct straight lines in a flat plane can do only one of two things. Either they run forever side by side and
never meet — we will call those parallel and study them in the next lessons — or they cross at
exactly one point. Not zero, not two: a single crossing. (Recall from Stage 13 that a line goes on without end in
both directions; two of them can only agree at one spot, or never.)
At that crossing the two lines slice the plane into four wedges — four angles meeting at the point. We name
the crossing point with a letter, say O (for the origin of the angles), and go around
it labelling the angles ∠1, ∠2, ∠3,
∠4. The labels just go around in order — like the hours on a clock.
One crossing point O, four angles. Going around: ∠1,
∠2, ∠3, ∠4.
Key idea
Two different lines in a plane either are parallel or meet at exactly one point. Where they meet, they make
four angles around the crossing point.
14.1.2 Neighbors on a line add to 180° (linear pairs)
Look at two angles that sit next to each other at the crossing — they share one side, and their two
outer sides lie along the same straight line. A pair like that is called a linear pair. Because the two
outer sides together form a perfectly straight line, the two angles fold open across a straight angle, and a
straight angle is 180°:
∠1 + ∠2 =
180°
This is exactly the supplementary idea from Stage 13 (two angles that add to 180°) — only now the two angles
are built into the picture, sitting back to back along the line. Go around the crossing and you will find
four linear pairs in all: ∠1+∠2, ∠2+∠3, ∠3+∠4, and ∠4+∠1 — each neighbor-pair stretched across a straight
edge.
A linear pair: ∠1 and ∠2 share the upward ray and their
other sides open out along one straight line, so they fill 180°.
From Stage 13
Two angles are supplementary when they add to 180°. A linear pair is just a supplementary pair you
can see in the figure — its outer sides make one straight line. Every linear pair is supplementary; that is
why ∠1 + ∠2 = 180°.
14.1.3 Vertical angles are equal
Now look at the two angles that sit directly opposite each other across the crossing — they do not
share a side, only the point O. These are vertical angles (sometimes "vertically
opposite angles"). The headline fact of this lesson is short and powerful:
Vertical angles are equal:∠1 = ∠3 and ∠2 = ∠4.
This is not something you have to measure and trust — you can reason it from the rule you already have.
Watch the two-line argument; this is geometry's first real "proof":
Why vertical angles are equal
∠1 + ∠2 = 180° (linear pair)
∠3 + ∠2 = 180° (linear pair)
Both ∠1 and ∠3 equal 180° − ∠2, the very same number — so ∠1 = ∠3. The same
argument run with ∠1 in the shared role gives ∠2 = ∠4. ∎
Notice how it worked: two different angles were each tied to the same third angle by a linear pair, so they
had to match each other. That little move — "both equal the same thing, so they equal each other" — will come back
again and again.
Try it Cross — turn one line and watch the family move
Tilt one line. The opposite (vertical) angles stay equal; each neighbor pair always sums to 180°.
tilt of the slanted line θ
emphasise
14.1.4 Reasoning around the point
Here is where the two rules pay off. Suppose somebody tells you just one of the four angles. You can now name
every other one — and, just as important, say why on each step. Let us do it with a number.
Worked example
Given ∠1 = 50°. Find ∠2, ∠3, and ∠4.
∠1 = 50° (given)
∠3 = 50° (vertical with ∠1)
∠2 = 130° (linear pair with ∠1: 180° − 50°)
∠4 = 130° (vertical with ∠2)
Every step names a rule. That is what turns an answer into reasoning.
Watch the pattern: the four angles came out as only two different sizes — a pair of 50°
and a pair of 130° — and those two sizes themselves add to 180°. That is always how a plain
crossing looks: two sizes, opposite-equal, neighbor-supplementary.
Try it Find all four — from one angle
Set ∠1. The widget fills in ∠2, ∠3, ∠4 and prints the reasoned chain, one "because" at a time.
∠1 =50°
Watch out
Do not mix up the pairings. Vertical angles are opposite — they share only the vertex
O, no side, and they are equal. Adjacent (linear-pair) angles are
neighbors — they share a whole side and add to 180°. Calling two side-by-side angles "vertical" is
the classic slip. Opposite = equal; neighbor = supplementary.
★ Recap
The pair
How they sit
The rule
Linear pair (adjacent)
neighbors — share a side, outer sides make a line
add to 180°
Vertical (opposite)
across the point — share only the vertex
are equal
One angle is enough. At a plain crossing the four angles are just two sizes:
θ and 180° − θ. Opposite angles are equal; neighbors sum to 180°.
Name one, reason out the rest, and write the rule on every line.
✎ Exercises
Work each one, then open the answer. Say the
rule out loud as you go — that is the real skill here.
Two lines cross at O, making ∠1, ∠2, ∠3, ∠4 in order around the point.
Which angle is the vertical-angle partner of ∠2?
Answer
∠4. Vertical angles sit directly opposite across the crossing, so ∠2 pairs with ∠4 (and ∠1
pairs with ∠3).
At a crossing, ∠1 = 70°. Find ∠2, ∠3, and ∠4, naming a rule for each.
Answer
∠3 = 70° (vertical with ∠1); ∠2 = 110°
(linear pair with ∠1: 180° − 70°); ∠4 = 110° (vertical with ∠2).
Two vertical angles are measured as 3x and 75°.
Find x.
Answer
Vertical angles are equal, so 3x = 75, giving x = 25. (Both angles are then 75°.)
Two adjacent angles on a straight line measure x and
2x. Solve for x.
Answer
They form a linear pair, so they add to 180°: x + 2x = 180 → 3x = 180 →
x = 60. (The angles are 60° and 120°.)
Lines AB and CD cross at O.
Reading around the point the rays come in the order A, C, B, D, so ∠AOC and ∠BOD sit opposite each other.
Are ∠AOC and ∠BOD a vertical pair or a linear pair?
Answer
A vertical pair. They sit opposite across O, sharing only the vertex —
so ∠AOC = ∠BOD. (A linear pair, like ∠AOC and ∠COB, would be neighbors sharing the
ray OC.)
Explain why vertical angles are equal, using only the linear-pair rule.
Answer
Let the vertical pair be ∠1 and ∠3, with ∠2 between them. Then ∠1 + ∠2 = 180° (linear pair) and
∠3 + ∠2 = 180° (linear pair). Both ∠1 and ∠3 equal 180° − ∠2, the same number, so ∠1 = ∠3.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson is where geometry turns from measuring into reasoning. The two facts — a linear pair
sums to 180°, and vertical angles are equal — are tiny, but the habit they build is the whole game: state the
given, name the rule, write the one-line conclusion. Encourage your student to say "because" on
every step ("∠3 = 50° because it's vertical with ∠1"). The vertical-angle proof in 14.1.3 is most students'
first genuine deductive argument; it is worth lingering on the move "both equal the same thing, so they equal each
other."
Misconception to watch: calling two adjacent (linear-pair) angles "vertical." Vertical angles are
opposite and share only the vertex; adjacent angles are neighbors and share a whole side. A quick
check: vertical angles are equal, adjacent angles on a line add to 180° — if a student says two side-by-side angles are
"equal because vertical," that is the slip.
Common Core:7.G.B.5 — use facts about supplementary, complementary, vertical, and
adjacent angles to write and solve simple equations for an unknown angle (exercises 3 and 4 do exactly this); and
HS G-CO.C.9 — prove that vertical angles are congruent (the two-line argument in 14.1.3).
eastmath.com · Stage 14 · 14.1 The Family of Angles · Reasoning, one step at a time
eastmath.com · 14.1 Two Lines Crossing: The Family of Angles · 14.1.3 Vertical angles are equal