Parallel lines, equal angles
Between parallel lines, corresponding angles are equal.
Will they match?
The lines l and m are parallel, and a transversal t crosses both. At the top crossing, ∠1 is 118°.
∠5 is the corresponding angle of ∠1, in the same position at the lower crossing. How big do you think ∠5 is?
What this lesson covers
The idea
Corresponding angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
Will they match?
The lines l and m are parallel, and a transversal t crosses both. At the top crossing, ∠1 is 118°.
∠5 is the corresponding angle of ∠1, in the same position at the lower crossing. How big do you think ∠5 is?
- Also 118°
- Smaller than 118°
- Bigger than 118°
Slide a tracing paper
A tracing paper has a copy of the four angles at the top crossing. Slide it down along t.
The paper fits
l and m are parallel, so they run in the same direction. Sliding the paper down t keeps its lines on the lines of the page. Each angle lands exactly on its corresponding angle, so the two are equal.
Corresponding angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
- At l | At m | Size
- ∠1 | ∠5 | 118°
- ∠2 | ∠6 | 62°
- ∠3 | ∠7 | 118°
- ∠4 | ∠8 | 62°
Notes
Corresponding angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
Check yourself
The lines l and m are parallel. ∠1 = 118°. Find ∠6.
Answer: 62 °
∠1 + ∠2 = 180°, so ∠2 = 180° − 118° = 62°. ∠6 corresponds to ∠2 and the lines are parallel, so ∠6 = 62°.
The lines l and m are parallel. Which statement is true?
The lines l and m are parallel. ∠4 = 70°. Find ∠5.
Answer: 110 °
∠1 + ∠4 = 180°, so ∠1 = 180° − 70° = 110°. ∠5 corresponds to ∠1, so ∠5 = 110°.
Why does the tracing paper fit exactly on the lower crossing?
- ∠3 = ∠6. ∠3 is at the bottom right of l and ∠6 at the top right of m. They are not corresponding angles, and they are not equal.
- ∠3 + ∠7 = 180°. Corresponding angles are equal, not supplementary: ∠3 + ∠7 would be twice ∠3.
- ∠3 = ∠7 — correct. ∠3 and ∠7 are corresponding angles, so for parallel lines they are equal.
- The lines are parallel, so they run in the same direction. — correct. Yes! The same direction means the lines on the paper land on the lines of the page.
- Because the lines are very long.. The length does not matter. What matters is the direction of the lines.
- Because the transversal is perpendicular to the lines.. It fits for any turn of the transversal. It is the parallel lines that make it fit.