Interior angles on the same side

The idea

The interior angles on the same side of a transversal intersecting a pair of parallel lines always add up to 180°.

Between the lines, same side

The lines l and m are parallel. ∠3 and ∠6 are on the same side of the transversal, between the lines. They are called interior angles on the same side.

What do you expect when we add ∠3 and ∠6?

• They add up to 180°
• They add up to 90°
• They are equal

Slide ∠6 up

Slide a copy of the lower angle up along t. Then turn t and switch sides.

A straight line

The copy of ∠6 lands on ∠2, because corresponding angles are equal. And ∠2 sits right next to ∠3 on a straight line, so together they make 180°.

The interior angles on the same side of a transversal intersecting a pair of parallel lines always add up to 180°.

• Step | Why
• ∠6 = ∠2 | corresponding angles, parallel lines
• ∠3 + ∠2 = 180° | linear pair
• ∠3 + ∠6 = 180° | interior angles on the same side

Notes

The interior angles on the same side of a transversal intersecting a pair of parallel lines always add up to 180°.

Check yourself

The lines l and m are parallel. ∠3 = 118°. Find ∠6.

Answer: 62 °

Interior angles on the same side add up to 180°. So ∠6 = 180° − 118° = 62°.

The lines l and m are parallel. ∠4 = 70°. Find ∠5.

Answer: 110 °

∠4 and ∠5 are interior angles on the same side, so ∠4 + ∠5 = 180°. ∠5 = 180° − 70° = 110°.

The lines l and m are parallel. In the picture, which two angles add up to 180° because they are interior angles on the same side?

The lines l and m are parallel. ∠1 = 75°. Find ∠6.

Answer: 105 °

∠1 + ∠2 = 180°, so ∠2 = 105°. ∠6 is the corresponding angle of ∠2, so ∠6 = 105°.

• ∠3 and ∠6 — correct. Yes! ∠3 = 118° and ∠6 = 62°, and 118° + 62° = 180°.
• ∠3 and ∠5. ∠3 and ∠5 are alternate angles, so they are equal: 118° each. Their sum is 236°, not 180°.
• ∠1 and ∠5. ∠1 and ∠5 are corresponding angles, so they are equal: 118° each. Their sum is 236°, not 180°.