Vertical angles (vertically opposite angles)

Vertical angles are the pairs of angles opposite each other where two straight lines cross. They are always equal, which is called the vertical angles theorem. Every crossing makes two pairs of vertical angles. In the UK they are called vertically opposite angles.

What are vertical angles?

Draw two straight lines that cross. They make four angles around the crossing point, numbered 1 to 4 here:

Angles 1 and 3 face each other across the crossing, so they are a pair of vertical angles. So are angles 2 and 4. The name has nothing to do with up and down: "vertical" comes from vertex, the point where the lines meet, because the two angles share that vertex. A pair can sit side by side, like 2 and 4, and still be vertical angles.

Vertical angles never share a side. Angles next to each other, such as 1 and 2, share a side and are called adjacent angles instead.

The vertical angles theorem

Here is why. Angles 1 and 2 sit together along one of the lines, so their total is 180°. Angles 2 and 3 sit together along the other line, so their total is 180° as well:

Both sums equal 180° and both contain angle 2, so whatever is left over must be the same: angle 1 = angle 3. The same argument with angle 1 in the middle shows angle 2 = angle 4. This proof is one of the theorems about lines and angles that the US Common Core standards ask high school students to prove.

Vertical angles and linear pairs

The proof uses a second fact you will need all the time. Two adjacent angles that make a straight line are called a linear pair, and they are supplementary: they add up to 180°. So at any crossing:

• Opposite angles are equal (vertical angles).
• Angles next to each other add up to 180° (a linear pair).
• All four angles add up to 360°, because they go all the way round a point.

That means one angle tells you all four. If one angle is 58°, its vertical angle is 58°, and the other two are each 180° − 58° = 122°. Check: 58 + 122 + 58 + 122 = 360.

If the two lines are perpendicular, all four angles are 90°, and each one is vertical to one right angle and adjacent to two others.

How to find x with vertical angles

Most questions give the angles as expressions in x and ask you to find x. Grade 7 in the US Common Core asks for exactly this: using facts about vertical, adjacent, supplementary and complementary angles to write and solve an equation for an unknown angle. The method is always the same:

1. Decide how the two angles are related. Opposite each other: vertical, so they are equal. Side by side on a straight line: a linear pair, so they add to 180°.
2. Write the equation. Equal angles: set the expressions equal. Linear pair: add them and set the sum to 180.
3. Solve for x, then put x back in to find the angle if the question asks for it.

Example 1: two vertical angles

Two vertical angles measure (3x + 10)° and (5x − 30)°. Find x and the size of each angle.

Vertical angles are equal, so set the expressions equal and solve:

So x = 20. Each angle is 3(20) + 10 = 70°, and the check 5(20) − 30 = 70° agrees.

Example 2: a linear pair

Two lines cross. Two angles next to each other are (x + 25)° and (2x − 10)°. Find all four angles.

These are adjacent on a straight line, so they add to 180°. Setting them equal would be the mistake here.

The two angles are 55 + 25 = 80° and 2(55) − 10 = 100°. Their vertical angles match them, so the four angles are 80°, 100°, 80° and 100°.

Example 3: three lines through one point

Three straight lines pass through the same point and make six angles. Going round, three neighbouring (neighboring in the US) angles are 50°, a and 60°. Find a.

Three neighbouring angles here fill exactly half a turn, because each of the other three is vertical to one of them. Half a turn is 180°, so a = 180 − 50 − 60 = 70°. The six angles are 50°, 70°, 60°, 50°, 70° and 60°, which add up to 360°.

Example 4: two unknowns

Two lines cross. One angle is 110°. The angle vertical to it is (2x + y)°, and the angle next to it is (3x − y)°. Find x and y.

The vertical angle equals 110°, and the adjacent angle is 180° − 110° = 70°. That gives two equations:

Add them and y cancels: 5x = 180, so x = 36. Then y = 110 − 72 = 38. Check: 3(36) − 38 = 70. The simultaneous equations calculator solves pairs like this step by step, and the Algebra course teaches the method.

Vertical angles in real life

Open a pair of scissors and the angle between the blades equals the angle between the handles: those are vertical angles. The same is true of an X shape, a railway crossing sign, two roads crossing in a straight line, or the legs of a folding chair. Open the scissors wider and both angles grow by the same amount.

Common mistakes

• Adding vertical angles to 180°. Vertical angles are equal. It is adjacent angles on a straight line that add to 180°.
• Using the rule when the lines are not straight. The theorem needs two straight lines crossing. If a line bends at the crossing point, the opposite angles need not be equal.
• Thinking vertical means up and down. It means sharing a vertex. Left and right pairs count too.
• Stopping at x. Read the question again: if it asks for the angle, put x back in.
• Trusting the diagram. Diagrams are often not drawn to scale. Use the facts, not a protractor.

To practise angle facts alongside other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Foundation.

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