Skip to content

Vertically opposite angles

Geometry · level 4, lesson 2 of 6

A question to start

Open a pair of scissors a little way. The two blades make a narrow angle at the front. Now look at the handles at the back: they make an angle too.

Open the scissors wider. The front angle gets bigger, and so does the back one. Close them, and both shrink together. Are the two angles always exactly the same size, or only roughly? And if they are exactly equal, how could you be sure without measuring every possible opening?

The idea

Two lines crossing

When two straight lines cross, they make four angles around the crossing point, like an X. Label them \(a\), \(b\), \(c\), \(d\) going round.

a b c d

Angles that are directly across the crossing point from each other, like \(a\) and \(c\), or \(b\) and \(d\), are called vertically opposite angles. ("Vertically" here has nothing to do with up and down. It comes from vertex: the two angles are opposite each other at the vertex. Some books just say "opposite angles".)

Notice that vertically opposite angles share the vertex but do not share an arm. Neighbouring angles like \(a\) and \(b\) share an arm and are adjacent.

The fact

Vertically opposite angles are equal. In the drawing, \(a = c\) and \(b = d\).

That is true however the lines cross: steep or shallow, nearly flat or nearly perpendicular. The scissors' front angle and back angle are always exactly equal.

Why it is true: a proof

Measuring a hundred X shapes would make us confident, but it could never cover every X. Instead, here is an argument that works for all of them at once, using only the fact from the last lesson.

a b a + b = 180° b c b + c = 180°
  1. Angles \(a\) and \(b\) sit side by side along one straight line (left drawing). So \(a + b = 180°\) (angles on a straight line).
  2. Angles \(b\) and \(c\) sit side by side along the other straight line (right drawing). So \(b + c = 180°\) (angles on a straight line).
  3. Both \(a\) and \(c\) make 180° together with the same angle \(b\). So \(a\) must be \(180° - b\), and \(c\) must be \(180° - b\) too.
  4. Therefore \(a = c\).

The same argument with \(b\) and \(d\) (both make 180° with \(c\)) shows \(b = d\).

That is a real proof: each step has a reason, and the reasons are facts we already trust. Because the argument never used any particular size of angle, it works for every pair of crossing lines. Nobody will ever find an X where it fails.

A second reason: a half turn

There is another way to see it. Put a pin through the crossing point and give the whole picture a half turn (180°). Each straight line turns into itself: the end that pointed up-right now points down-left, and the other way round. So after the half turn the picture looks exactly the same as before. But angle \(a\) has been carried to where angle \(c\) was. A turn does not change the size of an angle, so \(a = c\).

Two different reasons for the same fact is a good sign: it means we understand it, not just remember it.

What you get from one angle

Because of this fact and the straight-line fact, one angle tells you all four. If \(a = 110°\):

  • \(c = 110°\) (vertically opposite \(a\));
  • \(b = 180° - 110° = 70°\) (on a straight line with \(a\));
  • \(d = 70°\) (vertically opposite \(b\)).

The four angles always come in two equal pairs, and the two different sizes always add to 180°. If the lines are perpendicular, all four are 90°.

More than two lines

When three or more straight lines pass through the same point, every line is still straight, so the fact still works for any two angles that are directly opposite each other through the point.

40° 75° p q r s

Going round from the 40° angle, the six angles are 40°, 75°, \(p\), \(q\), \(r\), \(s\). Each angle is vertically opposite the angle three places further round, so \(q = 40°\), \(r = 75°\) and \(s = p\). And the first three angles fill a straight line, so \(p = 180 - 40 - 75 = 65°\). All six: 40°, 75°, 65°, 40°, 75°, 65°. Check: they add to \(2 \times 180 = 360\). ✓

Equations from opposite angles

When opposite angles are written with letters, setting them equal gives an equation.

3x + 10 5x − 30

The two marked angles are vertically opposite, so \(3x + 10 = 5x - 30\). Take \(3x\) from both sides: \(10 = 2x - 30\). Add 30 to both sides: \(40 = 2x\), so \(x = 20\). Each angle is \(3 \times 20 + 10 = 70°\), and the check is \(5 \times 20 - 30 = 70°\). ✓ The other two angles are \(180 - 70 = 110°\) each.

Notice the difference from the last lesson. Angles on a line give an equation that adds to 180. Opposite angles give an equation that says two things are equal. Knowing which fact fits the picture tells you which equation to write.

Worked examples

1. All four from one. Two lines cross and one angle is 38°. Find the other three.

  • The opposite angle is 38° (vertically opposite angles are equal).
  • The angles next to it are \(180 - 38 =\) 142° each (angles on a straight line; and they are opposite each other, so they agree).

Check: \(38 + 142 + 38 + 142 = 360\) (angles around a point). ✓

2. Opposite angles with letters. Two opposite angles are \(4y\) and \(y + 75\). Find \(y\).

\(4y = y + 75\) (vertically opposite). Take \(y\) from both sides: \(3y = 75\), so \(y = 25\). Each angle is \(4 \times 25 = 100°\). Check: \(25 + 75 = 100\). ✓

3. Adjacent angles with letters. Two adjacent angles at a crossing are \(2x\) and \(x + 15\). Find them.

Adjacent angles at a crossing lie on a straight line: \(2x + x + 15 = 180\), so \(3x = 165\) and \(x = 55\). The angles are \(110°\) and \(70°\). Check: \(110 + 70 = 180\). ✓

4. A road crossing. Two straight roads cross. The angle between them on the north side of the crossing is 65°. What is the angle between them on the south side, directly across the crossing? The two roads are straight lines, and the south angle is vertically opposite the north angle, so it is 65°. The angles on the east and west sides are \(180 - 65 = 115°\) each.

5. Why the lines must be straight. Suppose two "lines" cross but one of them is slightly bent at the crossing point, like a hinge. Then step 2 of the proof fails: the angles \(b\) and \(c\) no longer sit on a straight line, so they need not add to 180°, and \(a\) need not equal \(c\). A proof tells you not only that a fact is true, but when it is true.

Try it

  1. Two lines cross. One angle is 125°. Find the other three.
  2. Two lines cross. One angle is 90°. What are the others? What do we call these lines?
  3. Two vertically opposite angles are \(2x + 20\) and \(80\) (degrees). Find \(x\).
  4. Two vertically opposite angles are \(6t - 15\) and \(4t + 25\). Find \(t\) and the size of the angles.
  5. At a crossing, one angle is three times the angle next to it. Find all four angles.
  6. Three straight lines meet at a point, making six angles. Going round, three of them in a row are 50°, 60° and \(m\). Find \(m\) and the other three angles.
  7. Explain in your own words why vertically opposite angles are equal. Use the words "straight line" in your answer.
  8. In the scissors question at the start, why does it not matter how wide the scissors open?
Answers
  1. Opposite: 125°. Adjacent: \(180 - 125 =\) 55°, twice. (55°, 125°, 55°, 125°.)
  2. All four are 90°: the opposite one is 90°, and each neighbour is \(180 - 90 = 90°\). The lines are perpendicular.
  3. \(2x + 20 = 80\), so \(2x = 60\) and \(x = 30\).
  4. \(6t - 15 = 4t + 25\), so \(2t = 40\) and \(t = 20\). Each angle is \(6 \times 20 - 15 = 105°\) (check: \(4 \times 20 + 25 = 105\) ✓).
  5. Call the smaller one \(x\); the one next to it is \(3x\). They are on a straight line: \(x + 3x = 180\), so \(4x = 180\) and \(x = 45\). The angles are 45°, 135°, 45°, 135°.
  6. Three neighbouring angles of the six fill a straight line (half the way round), so \(m = 180 - 50 - 60 =\) 70°. The other three are opposite these: 50°, 60°, 70°.
  7. For example: "Each of the two opposite angles sits on a straight line with the same angle between them, so each one is 180° minus that angle. So they are equal."
  8. The blades and handles are two straight pieces crossing at the screw. Whatever the opening, the front angle and the back angle are vertically opposite, and the proof works for every opening.

Common mistake

Calling the wrong pair "opposite". Vertically opposite angles are across the vertex from each other, with no shared arm. Two angles next to each other (sharing an arm) are adjacent, and they add to 180°, not equal each other (unless both are 90°).

Using the fact when a line is not straight. The proof needs both lines to be straight through the crossing point. If a line bends there, or if two separate lines just touch at their ends, the fact does not apply.

Where it leads