Math Resources Hub

Intersecting Lines

When two straight lines cross, four angles appear at the meeting point — and they are not all different. They fall into two matching pairs, no matter how you tilt the lines.

55°55°125°125°
Wherever two lines cross — the blades of a pair of scissors, a railroad crossing, a plain X — the four angles come in two equal pairs.

Explore

Drag the slider to change the angle between the lines. Click an angle to select it, and switch between the two views. Watch for the pattern: which angles stay equal to each other, and which pair up to make a straight line?

60°

Click an angle to choose it, drag the slider to rotate the second line, then press “Show why”.

The selected angle and its opposite are a vertical pair — 60° and 60° — and vertical angles are always equal. The other pair each measure 120°.

Two lines crossing, with the angle between them 60°. The selected angle and its opposite (single-arc) form a vertical pair, each 60°; the other pair (double-arc) each measure 120°. Vertical angles are always equal — a half-turn about the crossing maps one exactly onto the other.

Opposite angles

Make a prediction: Two lines cross. One of the four angles measures 110°. What is the angle directly opposite it — the one sharing only the crossing point?

Angles on opposite sides of the crossing are , and vertical angles are always (equal in measure). In the diagram, the two orange angles are equal to each other, and so are the two blue ones.

Neighboring angles

Make a prediction: An angle of 110° sits next to a neighboring angle, and together their outer sides form one straight line. What must the two angles add up to?

Two angles that sit next to each other along a straight line form a . Their outer sides make a straight angle, so a linear pair is always — the measures add to 180°.

55°125°55° + 125° = 180°
A linear pair: the orange angle and its violet neighbor share a side, their outer sides form a straight line, and together they make 180°.

Test it

Go back to the tool and drag the slider through many different tilts. Try to find any crossing where the two opposite angles are not equal, or any neighboring pair that does not total 180°. You won't — and that is what makes these relationships theorems rather than coincidences.

Why it works

The two facts are linked: the linear-pair rule forces the vertical-angle rule.

Pick one of the orange angles, 55°. Each of its two neighbors forms a linear pair with it, so each neighbor is 180°−55°=125°180° - 55° = 125° — the blue angles. Now the angle opposite the orange one is itself a neighbor of a blue angle, so it is 180°−125°=55°180° - 125° = 55° — orange again. Opposite angles have to match because they are each "180° minus the same neighbor."

Key result

When two lines cross, vertical angles are congruent and each linear pair is supplementary (adds to 180°). Know one of the four angles and you know all four.

A special case is worth a look: when the lines cross at a right angle, every one of the four angles is 90°.

90°90°90°90°
Perpendicular lines. One angle is 90°, so its linear-pair neighbors are 180° − 90° = 90°, and the vertical angle is 90° too — all four are right angles.
Worked example: Two lines cross and one angle measures 35°. Find the other three.

  1. The angle straight across from the 35° angle is its vertical angle, so it is also 35°.

    Before the next step: What is the measure of the angle directly opposite the 35° angle?

Practice

Score: 0/4

  1. 1.Two lines cross. One angle measures 72°. What is its vertical angle (the one directly opposite)?
  2. 2.Two angles form a linear pair. One measures 130°. What is the measure of the other, in degrees?
    °
  3. 3.Two lines cross so that one of the four angles is a right angle (90°). What are the other three angles?
  4. 4.Two vertical angles measure (2x + 10)° and (x + 40)°. Find x.
For teachers

Standard, common misconceptions, and suggested use will live here.