Alternate angles are equal

The idea

Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal, since an angle equals its corresponding angle, which equals its own vertically opposite angle.

Will the alternate angle match?

The lines l and m are parallel. ∠6 is 62°. ∠4 is its alternate angle, on the other side of the transversal.

How big do you think ∠4 is?

• Also 62°
• Smaller than 62°
• Bigger than 62°

Hop and watch the size

The lines are parallel, so the sizes show. Make the two hops from the lit angle and watch the size.

Equal at every hop

Each hop keeps the size. Hop 1 goes to a corresponding angle: equal, because the lines are parallel. Hop 2 goes to a vertically opposite angle: always equal. So the alternate angle is equal too.

Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal, since an angle equals its corresponding angle, which equals its own vertically opposite angle.

• Step | Why equal
• ∠6 = ∠2 | corresponding angles, parallel lines
• ∠2 = ∠4 | vertically opposite angles
• So ∠6 = ∠4 | alternate angles

Notes

Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal, since an angle equals its corresponding angle, which equals its own vertically opposite angle.

Check yourself

The lines l and m are parallel. ∠1 = 118°. Find ∠6, the alternate angle of ∠4.

Answer: 62 °

∠1 + ∠4 = 180°, so ∠4 = 62°. Alternate angles between parallel lines are equal, so ∠6 = ∠4 = 62°.

The lines l and m are parallel. In the picture, which two angles are equal because they are alternate angles?

The lines l and m are parallel. Why is ∠6 = ∠4?

The lines l and m are parallel. ∠2 = 130°. Find ∠5, the alternate angle of ∠3.

Answer: 50 °

∠2 + ∠3 = 180°, so ∠3 = 50°. Alternate angles between parallel lines are equal, so ∠5 = ∠3 = 50°.

• ∠3 and ∠6. ∠3 and ∠6 are not alternate angles. In the picture one is 118° and the other 62°.
• ∠4 and ∠6 — correct. Yes! ∠6 corresponds to ∠2, and ∠2 is vertically opposite to ∠4, so ∠4 = ∠6.
• ∠1 and ∠6. ∠1 and ∠6 are not alternate angles. In the picture one is 118° and the other 62°.
• Because both are between the lines.. Being between the lines is only where they are. The equality comes from corresponding and vertically opposite angles.
• Because ∠6 + ∠4 = 180°.. ∠6 and ∠4 are not a linear pair. They are alternate angles, and they are equal.
• ∠6 = ∠2 (corresponding angles) and ∠2 = ∠4 (vertically opposite angles). — correct. Yes! Two equalities in a row: ∠6 = ∠2 = ∠4.