Adjacent and vertically opposite angles

Angles that work themselves out

Two lines cross — like scissors, or two paths on a map. At the crossing four angles appear. And now the best bit: it is enough to measure one and you work out the other three in your head.

Today’s lesson stands on two rules:

Adjacent angles — two angles that sit next to each other on one straight line (they share an arm and the remaining arms make a straight line). Together they always make 180°, because together they fill a straight angle. You know one → the other is 180 minus the first.

Vertically opposite angles — two angles opposite each other at a crossing (they only touch at the vertex). They are always the same size. That is why open scissors open the same at the top as at the bottom.

This is the first lesson where you do not spot angles by measuring, but by calculating. Geometry stops being about putting a protractor on and starts being about thinking — and that is exactly the direction we swim further.

Sofie and Adam look at a map where a field path crosses a road. “The path turns off the road at an angle of 40°,” Sofie reads. Adam puts a protractor on the other side of the crossing: “And here I get… hang on… 140. That’s luck!” Sofie shakes her head: “No luck. Those two angles sit on one straight line — the road. Together they must make 180. One hundred and eighty minus forty is one hundred and forty.” Adam measures the other two angles of the crossing too: 40° and 140°. Four angles, and yet only two different numbers. A crossing has no secrets.

Two rules and how to count with them

Rule 1: Adjacent angles add to 180°.

Why: their outer arms make a straight line and a straight line is a straight angle, 180°. Two adjacent angles split those 180° between them.

Calculation: the other adjacent angle = 180° − the known angle. Next to 40° the adjacent is 140°. Next to 90° the adjacent is 90° again — a right angle has a right neighbour (that is why door corners are right from every side).

Rule 2: Vertically opposite angles are equal.

Why: both are adjacent to the same angle. If angle α is adjacent to β, then α = 180° − β. And the opposite angle γ is also adjacent to β, so γ = 180° − β. Two same subtractions → the same result → α = γ. No measuring, pure logic.

Method at a crossing:

1. Find the given angle.
2. The angle opposite (across the vertex) = the same.
3. Both angles next to it (on the line) = 180° minus the given one.

A check at the end: the sum of all four angles around the vertex must be 360° — a full angle. 40 + 140 + 40 + 140 = 360 ✓. If it does not sit, there is a mistake somewhere.

Work out a crossing with 65°

Two lines cross, one angle measures 65°. The other three:

Vertically opposite (across the vertex): the same → 65°.

Adjacent (neighbour on the line): 180 − 65 = 115°.

The fourth (vertically opposite to the adjacent): 115°.

Check with a full angle: 65 + 115 + 65 + 115 = 360 ✓.

One measurement, three calculations, four angles. Faster than putting a protractor on three times — and more precise.

Find the adjacent angle to 90° and to 132°

To a right angle (90°): 180 − 90 = 90°. The adjacent angle of a right angle is right again! So when one corner of a crossing is right, all four are right — that is how a right-angled street crossing is made.

To an obtuse angle of 132°: 180 − 132 = 48°. The adjacent angle of an obtuse angle is always acute — together they must make a straight line and the obtuse one took more than half of it.

Scissors: why they open the same at the top and the bottom

You open scissors so the blades make 30°. What angle do the handles make?

The blades and the handles sit on the same two lines, which cross at the screw — the vertex. The angle of the blades and the angle of the handles are vertically opposite: they sit opposite each other across the vertex.

Vertically opposite angles are equal → the handles make 30° too.

That is why scissors “obey”: however much you open the handles, the blades open by exactly that much. Vertically opposite angles in action, a hundred times a day.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Work out adjacent angles: 180 minus the given angle. Write the answer as a number (degrees with no mark).

Practise the questions

On paper

Draw crossings and work them out — use a protractor only to check.

1. Draw two lines crossing at about 50°. Measure one angle and work out the other three. Then measure all of them with a protractor — do they sit?

2. Work out in your head the adjacent angles to: 30°, 75°, 118°, 155°. Write a check by adding too (they must make 180°).

3. At a crossing one angle is 3× bigger than its adjacent angle. What are both? (Hint: together they make 180°, so 4 equal parts.)

4. Find a crossing at home (scissors, crossed pencils, a window grid) and for each one find all four angles from a single measurement.

Now you 💪

1. I can work out an adjacent angle as 180 minus the given one — in my head.
2. I know that vertically opposite angles are equal, and I can say why.
3. I can check a crossing by adding: four angles around the vertex = 360°.

What to take from this lesson

• Adjacent angles sit on a straight line and add to 180°.
• Vertically opposite angles (opposite across the vertex) are always equal.
• At a crossing of two lines there are only two different angle values.
• Check: the sum of angles around a vertex is 360°.
• Angles can be calculated, not only measured — a calculation is more precise than a protractor.

© 2026 Ing. Martin Polak / AlgoRhino · 内容使用条款