One line across two lines

The idea

A line that intersects two lines is called a transversal; it forms eight angles which, since vertically opposite angles are equal, have at most four distinct measures.

A road across two streets

Picture a straight road that cuts across two other streets. Where it crosses each street, angles are made.

The road crosses the first street and then the second street. How many angles are made in all?

• 4
• 6
• 8

Open the eight angles

Line t crosses the lines l and m. Open the angles to see their sizes.

A transversal

A line that crosses two lines is called a transversal. Here t is the transversal. It makes 8 angles, 4 at each crossing. At each crossing the vertically opposite angles are equal, so the 8 angles show at most 4 different sizes.

A line that intersects two lines is called a transversal. It forms eight angles, and since vertically opposite angles are equal, they have at most four different measures.

• Crossing | Angles | Different sizes
• On l | 1, 2, 3, 4 | at most 2
• On m | 5, 6, 7, 8 | at most 2
• Both | 8 angles | at most 4

Notes

A line that intersects two lines is called a transversal. It forms eight angles, and since vertically opposite angles are equal, they have at most four different measures.

Check yourself

Which line is the transversal in the picture?

Why do the eight angles of a transversal show at most four different sizes?

At the crossing on line l, ∠1 = 118°. How many different sizes do the four angles at this crossing have?

Answer: 2

∠3 = ∠1 = 118°. ∠2 = 180° − 118° = 62°, and ∠4 = ∠2 = 62°. So there are 2 different sizes: 118° and 62°.

A transversal crosses two lines. At l the angles are 118° and 62°. At m the angles are also 118° and 62°. How many different sizes are there in all, among the eight angles?

Answer: 2

Only two sizes appear: 118° and 62°. The most possible is 4, but here the sizes at m repeat those at l.

• Line a. Line a is one of the two lines that are crossed. The transversal crosses them.
• Line b. Line b is one of the two lines that are crossed. The transversal crosses them.
• Line c, because it crosses both of the other lines — correct. Yes! A transversal is the line that crosses two other lines.
• At each crossing the vertically opposite angles are equal, so each crossing gives at most two sizes. — correct. Yes! Two crossings with at most two sizes each make at most four sizes.
• Because the transversal is always a straight line.. The transversal is straight, but that is not what limits the number of sizes. Think about each crossing.
• Because the two lines are always parallel.. The two lines need not be parallel. The limit comes from vertically opposite angles at each crossing.