Alternate angles

The idea

The alternate angle of an angle formed by a transversal is found by first taking its corresponding angle and then the angle vertically opposite to that corresponding angle.

From ∠6 to ∠4

∠6 and ∠4 sit on opposite sides of the transversal t, between the lines. They are called alternate angles. How can we get from ∠6 to ∠4 using what we already know?

Which way gets us from ∠6 to ∠4?

• Hop to the corresponding angle, then to the vertically opposite angle
• Measure both with a protractor
• Hop to the linear pair twice

Two hops

Start at the lit angle. Hop 1: its corresponding angle. Hop 2: the angle vertically opposite to where you landed.

The alternate angle

Every time, the two hops land on the alternate angle: the angle on the other side of the transversal, at the other line.

The alternate angle of an angle is found by first taking its corresponding angle, and then the angle vertically opposite to that corresponding angle.

• Start | Hop 1: corresponding | Hop 2: vertically opposite
• ∠6 | ∠2 | ∠4
• ∠3 | ∠7 | ∠5
• ∠8 | ∠4 | ∠2

Notes

The alternate angle of an angle is found by first taking its corresponding angle, and then the angle vertically opposite to that corresponding angle.

Check yourself

What is the alternate angle of ∠5?

To find the alternate angle of ∠8, which two hops do we make?

Are ∠2 and ∠8 alternate angles?

The alternate angle of ∠4 is ∠n. What is n?

Answer: 6

The corresponding angle of ∠4 is ∠8. The angle vertically opposite to ∠8 is ∠6. So the alternate angle of ∠4 is ∠6.

• ∠1. ∠1 is the corresponding angle of ∠5. One more hop: vertically opposite.
• ∠3 — correct. Yes! The corresponding angle of ∠5 is ∠1, and the angle vertically opposite to ∠1 is ∠3.
• ∠7. ∠7 is vertically opposite to ∠5. But the alternate angle starts with the corresponding angle.
• ∠4 first (corresponding), then ∠2 (vertically opposite) — correct. Yes! The alternate angle of ∠8 is ∠2.
• ∠4 first (corresponding), then ∠1 (linear pair). The second hop must go to the angle vertically opposite, not to a linear pair.
• ∠7 first (linear pair), then ∠3 (corresponding). The hops are corresponding and vertically opposite. A linear pair is a different relation.
• Yes. ∠8 corresponds to ∠4, and ∠4 is vertically opposite to ∠2. — correct. Yes! Two hops from ∠8 land on ∠2.
• No. They are corresponding angles.. The corresponding angle of ∠2 is ∠6, not ∠8.
• No. They are vertically opposite angles.. Vertically opposite angles share the same crossing point. ∠2 and ∠8 are at different lines.