Adjacent angles

Adjacent angles are two angles that share a vertex (corner point) and a side (arm), and don't overlap. In other words, they sit right next to each other, back to back. Together they make one bigger angle, so you can add them: that is what makes adjacent angles useful for finding missing angles.

What are adjacent angles?

Here three rays start from the point O, going through A, B and C:

Angle AOB and angle BOC are adjacent. They have the same vertex, O. They share the side OB. And they don't overlap: one is on each side of OB.

Angle AOB and angle AOC are not adjacent, even though they share the vertex O and the side OA. Angle AOC contains all of angle AOB inside it, so they overlap.

The three conditions

1. A common vertex: both angles have their corner at the same point.
2. A common side: one arm belongs to both angles.
3. No overlap: the angles lie on opposite sides of the shared arm, so no point is inside both.

All three must be true. Most "is this adjacent?" questions are testing which one fails.

Adjacent angles: examples and non-examples

At a crossing of two straight lines, angles next to each other are adjacent, while angles opposite each other are vertical angles, which share only the vertex. Two angles of a triangle share a side but have different vertices, so they are not adjacent either.

Adding adjacent angles

Because adjacent angles fit together with no gap and no overlap, the two of them make the outer angle exactly. US geometry calls this the angle addition postulate:

So if you know any two of the three angles, you can find the third by adding or subtracting. If OB cuts angle AOC into two equal adjacent angles, OB is called the angle bisector.

Special pairs of adjacent angles

When the outer angle is one you know, adjacent angles give you an equation straight away. The full list of angle facts is in the angles guide.

Complementary and supplementary angles don't have to be adjacent: any two angles with the right total count. But when they are adjacent, you can see the right angle or the straight line they make, which is how you know to use 90° or 180°.

Worked examples

In the US Common Core, grade 7 students use facts about adjacent, vertical, complementary and supplementary angles to write and solve equations for unknown angles, often in several steps. These examples do exactly that.

Example 1: angle addition

Angle AOC is 75° and angle AOB is 28°. Angles AOB and BOC are adjacent. Find angle BOC.

The two adjacent angles make angle AOC, so angle BOC = 75 − 28 = 47°.

Example 2: adjacent angles in a right angle

A right angle is split into two adjacent angles of (2x + 6)° and (3x − 1)°. Find x and both angles.

So x = 17, and the angles are 2(17) + 6 = 40° and 3(17) − 1 = 50°.

Example 3: a linear pair

Two adjacent angles form a straight line. One is five times the other. Call the smaller one x:

The angles are 30° and 150°.

Example 4: angles around a point

Four adjacent angles fill the space around a point. They are 110°, 2x, 3x and 50°. Find x.

So x = 40, and the two unknown angles are 80° and 120°. Check: 110 + 80 + 120 + 50 = 360.

Example 5: an angle bisector

OB bisects angle AOC, which is 96°. The two adjacent angles are equal, so each is 96 ÷ 2 = 48°. Equations like these are practised in the Algebra course.

Adjacent angles in shapes

In a polygon, "adjacent angles" usually means the angles at two neighbouring (neighboring in the US) corners, joined by one side. These are not adjacent in the strict sense above, because their vertices are different, so some textbooks call them consecutive angles instead. In a parallelogram, any two of these neighbouring angles add up to 180°. In a rectangle, they are both 90°.

Common mistakes

• Calling vertical angles adjacent. Opposite angles at a crossing share only a point, not a side.
• Missing the overlap. A small angle inside a bigger one with the same vertex and arm is not adjacent to it.
• Assuming adjacent angles add to 180°. Only if they make a straight line. Adjacent angles can add up to anything less than 360°.
• Assuming a line is straight. Use 180° only when the diagram or question says the line is straight.

To practise angle facts mixed with 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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