Corresponding angles

Corresponding angles are angles in the same position at two different crossings, when one line (the transversal) cuts across two other lines. If those two lines are parallel, corresponding angles are equal. That fact lets you find every angle in the diagram from just one.

What are corresponding angles?

Here line t crosses lines a and b. It makes four angles at each crossing, eight in total:

Two angles correspond when they sit in the same spot at their own crossing: above or below the line they are on, and on the same side of t. Angle 1 is above a and left of t. Angle 5 is above b and left of t. So 1 and 5 are corresponding angles. There are four pairs:

A quick way to spot a pair is to slide one crossing along t until it lands on the other. The angles that land on top of each other correspond. Some people trace an F shape through the pair, but that is only a way to find them, not a reason to write in a proof or exam.

The corresponding angles theorem

This makes sense from the sliding picture: parallel lines point the same way, so each crossing is an exact copy of the other. Many US geometry textbooks take it as a postulate, a fact you accept without proof, and then use it to prove the other parallel line facts. The Common Core standards list it among the theorems about lines and angles, and grade 8 students meet it through informal arguments about lines cut by a transversal.

In the diagram, if a and b are parallel, then angle 1 = angle 5, angle 2 = angle 6, angle 3 = angle 7 and angle 4 = angle 8.

The converse: proving lines are parallel

The theorem also works backwards. If a pair of corresponding angles are equal, the two lines are parallel. So to show two lines are parallel, measure or calculate one pair of corresponding angles. Equal means parallel. Different, even by one degree, means the lines will meet somewhere.

When the lines are not parallel

Corresponding angles still exist when a and b are not parallel, because the name only describes position. They just are not equal any more, so you can't use them to find missing angles.

Corresponding, alternate and other angle pairs

Corresponding angles are one of several pairs in this diagram. With parallel lines, all the angles are one of two sizes: here the odd-numbered angles are equal to each other, the even-numbered angles are equal to each other, and an odd angle plus an even angle always makes 180°.

Vertical angles and linear pairs work at any crossing. Corresponding, alternate and same-side interior angles only give you equal or supplementary angles when the lines are parallel. More on those: alternate interior angles, vertical angles and supplementary angles. For every angle fact in one place, see angles.

Worked examples

Example 1: find all eight angles

In the diagram, a and b are parallel and angle 1 is 65°. Find the other seven angles.

• Angle 5 = 65°, because corresponding angles are equal.
• Angle 3 = 65° and angle 7 = 65°, because vertical angles are equal.
• Angle 2 = 180 − 65 = 115°, because angles 1 and 2 are a linear pair.
• Angles 4, 6 and 8 are also 115°, by the same reasons.

So the odd angles are all 65° and the even angles are all 115°.

Example 2: solve for x

Lines a and b are parallel. Angle 2 is (4x − 15)° and angle 6 is (2x + 37)°. Find x and the angle.

Angles 2 and 6 are corresponding, so they are equal:

So x = 26 and each angle is 4(26) − 15 = 89°.

Example 3: two steps

Lines a and b are parallel. Angle 2 is (5x + 10)° and angle 5 is (3x + 10)°. Find x.

Angles 2 and 5 are not a named pair, so go through a third angle. Angle 2 corresponds to angle 6, and angles 5 and 6 are a linear pair. So angle 2 and angle 5 add up to 180°:

So x = 20, angle 2 is 110° and angle 5 is 70°. This is the same-side interior angles fact in disguise.

Example 4: are the lines parallel?

A transversal makes an angle of 72° with one line and, in the corresponding position, 74° with another. The corresponding angles are not equal, so the lines are not parallel. They lean towards each other by 2° and will meet on one side.

Corresponding angles in similar and congruent shapes

The phrase has a second meaning. When two shapes are similar or congruent, the angles that match up are also called corresponding angles, and they are always equal. If triangle ABC is similar to triangle DEF, then angle A = angle D, angle B = angle E and angle C = angle F. The order of the letters tells you which angles correspond. Enlarging a shape changes its side lengths but never its angles.

Common mistakes

• Assuming lines are parallel. Only use "corresponding angles are equal" if the lines are marked with arrows or stated to be parallel.
• Mixing up corresponding and alternate. Corresponding angles are on the same side of the transversal. Alternate angles are on opposite sides.
• Adding corresponding angles to 180°. They are equal. It is same-side interior angles that add up to 180°.
• Writing "F angles" as a reason. Use the proper name in your working.

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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Maths · GeometryVertical angles: definition, theorem and how to find xVertical angles are the opposite angles made when two lines cross, and they are always equal. See why, then solve for x with worked examples.