Alternate interior angles

Alternate interior angles are two angles that sit between a pair of lines, on opposite sides of a third line (the transversal) that crosses them both. When the pair of lines are parallel, alternate interior angles are equal. In the UK they are simply called alternate angles.

What are alternate interior angles?

Here the transversal t crosses lines a and b, making eight angles:

The name tells you where to look. Interior means between lines a and b, so only angles 3, 4, 5 and 6 qualify. Alternate means on opposite sides of t. That leaves two pairs:

• Angles 3 and 5: angle 3 is right of t, angle 5 is left of t.
• Angles 4 and 6: angle 4 is left of t, angle 6 is right of t.

Join the two angles of a pair with the lines around them and you trace a Z shape, or a backwards Z. That is a good way to spot them, but write the proper name in your working.

The alternate interior angles theorem

The US Common Core standards list this among the theorems about lines and angles that high school students prove. Here is the proof for angles 3 and 5, using two facts you already have:

1. Angle 3 = angle 1, because they are vertical angles.
2. Angle 1 = angle 5, because they are corresponding angles and a is parallel to b.
3. So angle 3 = angle 5.

The same steps with angles 2, 4 and 6 show that angle 4 = angle 6.

The converse: proving lines are parallel

The theorem works backwards too. If a transversal makes alternate interior angles that are equal, the two lines are parallel. If angle 3 is 61° and angle 5 is 63°, the lines are not parallel, however close they look.

Alternate exterior and same-side interior angles

Two related pairs often appear in the same questions, so it is worth knowing how they differ.

Same-side interior angles are also called consecutive interior angles in the US and co-interior angles in the UK. Because they add up to 180°, they are supplementary. Every other angle fact is collected in the angles guide.

Worked examples

Example 1: find two angles

In the diagram, a is parallel to b and angle 4 is 128°. Find angles 6 and 5.

• Angle 6 = 128°, because alternate interior angles are equal.
• Angle 5 = 180 − 128 = 52°, because angles 5 and 6 sit together on line b.

Example 2: solve for x

Lines a and b are parallel. Angle 3 is (3x + 17)° and angle 5 is (5x − 25)°. Find x and the angle.

So x = 21, and each angle is 3(21) + 17 = 80°.

Example 3: why a triangle's angles add up to 180°

Alternate angles give the neatest proof of the triangle angle sum, which grade 8 in the US Common Core explores. Take triangle ABC and draw a line through C parallel to AB.

1. At C, the new line makes two angles either side of the triangle's own angle C.
2. The angle on the left equals angle A, because they are alternate interior angles between the parallel lines, with AC as the transversal.
3. The angle on the right equals angle B, the same way with BC as the transversal.
4. The three angles at C fill a straight line, giving a total of 180°. So A + B + C = 180°.

For a triangle with base angles of 70° and 45°, the top angle is 180 − 70 − 45 = 65°.

Example 4: a bend between parallel lines

Two parallel lines run left to right. A path starts at A on the top line, runs down to a point P between the lines, then turns back and runs to B on the bottom line, making an arrowhead that points right. The angle between AP and the top line, on the right of A, is 35°. The angle between BP and the bottom line, on the right of B, is 50°. Find angle APB.

Draw a third parallel line through P. It splits the angle at P into two parts. The top part is an alternate angle to the 35° angle, and the bottom part is an alternate angle to the 50° angle. So angle APB is 35 + 50 = 85°. Adding a parallel line through a corner is the standard trick for any zigzag between parallel lines.

Common mistakes

• Using the theorem without parallel lines. Check for the arrow marks or the word parallel first.
• Picking an exterior angle. Interior means between the two lines. Angles 1, 2, 7 and 8 are exterior.
• Picking the same side. Angles 4 and 5 are both between the lines but on the same side of t, so they add up to 180°. They are not equal.
• Writing "Z angles". AQA's GCSE specification says colloquial terms such as Z angles are not acceptable. Write "alternate angles are equal".

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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