When two lines intersect, they create four angles at a single point. Using linear pairs and vertical angles lets you find all four angle measures from just one angle.

Key ideas (in simple words)

  • Linear pair :
    Two angles that:

    • Share a side (are adjacent), and
    • Their other sides form a straight line.
      So they form a straight angle and their measures add to 180° (they are supplementary).

  • Vertical angles :
    Two angles that:

    • Are opposite each other when lines cross,
    • Do not share a side.
      Vertical angles are always equal in measure.

Step‑by‑step: what happens when lines intersect

Imagine two lines crossing and forming angles 1, 2, 3, and 4 around the intersection.

Think of them going around a point: ∠1, ∠2, ∠3, ∠4 in order.

1. Use linear pairs (sum to 180°)

Each angle has two neighbors that form a straight line:

  • ∠1 and ∠2 form a linear pair → m∠1+m∠2=180°m∠1+m∠2=180°m∠1+m∠2=180°.
  • ∠2 and ∠3 form a linear pair → m∠2+m∠3=180°m∠2+m∠3=180°m∠2+m∠3=180°.
  • ∠3 and ∠4 form a linear pair → m∠3+m∠4=180°m∠3+m∠4=180°m∠3+m∠4=180°.
  • ∠4 and ∠1 form a linear pair → m∠4+m∠1=180°m∠4+m∠1=180°m∠4+m∠1=180°.

So any angle plus the one next to it on a straight line must add to 180°.
This means: once you know one angle, you can find its adjacent angle by doing
“180° − (known angle).”

2. Use vertical angles (congruent/opposite)

Now look at opposite angles:

  • ∠1 and ∠3 are vertical → m∠1=m∠3m∠1=m∠3m∠1=m∠3.
  • ∠2 and ∠4 are vertical → m∠2=m∠4m∠2=m∠4m∠2=m∠4.

So each angle has a twin directly across from it with the same measure.

Putting it together with an example

Say you are told:

  • m∠1=70°m∠1=70°m∠1=70°.

Use the properties:

  1. Vertical angles :

    • ∠3 is vertical to ∠1 → m∠3=70°m∠3=70°m∠3=70°.

  2. Linear pair with ∠1 :

    • ∠1 and ∠2 form a linear pair, so
      m∠1+m∠2=180°m∠1+m∠2=180°m∠1+m∠2=180°
      70°+m∠2=180°70°+m∠2=180°70°+m∠2=180°
      m∠2=110°m∠2=110°m∠2=110°.

  3. Vertical pair with ∠2 :

    • ∠2 and ∠4 are vertical → m∠4=110°m∠4=110°m∠4=110°.

Result:

  • ∠1 = 70°, ∠3 = 70° (vertical angles),
  • ∠2 = 110°, ∠4 = 110° (vertical angles),
  • Each adjacent pair adds to 180° (linear pairs).

Final explanation in one go

When two lines intersect, each angle has:

  • An opposite angle that is equal to it (vertical angles), and
  • Two adjacent angles that are supplementary to it (linear pairs, sum 180°).

So if you know the measure of just one of the four angles, you can:

  • Use vertical angles to get the opposite angle (same measure), and
  • Use linear pairs to find the two neighboring angles (180° minus the known angle).

This combination guarantees you can determine all angle measures formed by the intersecting lines. TL;DR:
Vertical angles give you equal opposite angles.
Linear pairs give you supplementary adjacent angles (add to 180°).
Together, they let you find every angle created when two lines intersect from just one given angle.