Math Resources Hub

Exterior Angle

Push one side of a triangle straight past a corner, like extending a road beyond an intersection. The angle that opens up outside the triangle is an — and it carries a surprising piece of information about the two corners far away from it.

70°50°120°ABCD120° = 70° + 50°
Extend side BC past C and the exterior angle (120°) opens up. It exactly equals the two remote interior angles added together: 70° + 50°.

Explore

Reshape the triangle and watch the exterior angle at the extended corner. Keep an eye on the two remote interior angles — the two corners that are not next to it. What relationship holds between the exterior angle and those two?

Reshape the triangle with the sliders — or drag vertex B or C — and watch the exterior angle stay equal to ∠A + ∠C. Then press “Show why”.

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45°

The exterior angle at B is 91°, equal to the sum of the two remote interior angles: ∠A = 45° plus ∠C = 46° is 91°.

A triangle with the base AB extended past B. The exterior angle there is 91°, which always equals the sum of the two remote interior angles, ∠A = 45° and ∠C = 46° (45° + 46° = 91°). Press “Show why” to see the proof: a ray from B parallel to side AC splits the exterior angle into a copy of ∠C and a copy of ∠A, which tile it exactly.

Make a prediction

Make a prediction: A triangle's two remote interior angles measure 70° and 45°. What is the exterior angle at the third vertex?

Test it

In the tool, notice two facts holding at once. The exterior angle and the interior angle right beside it always make a straight line — they are a linear pair, summing to 180°180°. And the exterior angle always equals the two combined. Try to find a triangle where either rule fails. You can't.

Why it works

Both facts come straight from the angle sum. Call the interior angles AA, BB, and CC, and put the exterior angle at CC. The exterior angle and CC form a linear pair, so:

exterior=180°−C\text{exterior} = 180° - C

But the three interior angles also total 180°180°, so 180°−C=A+B180° - C = A + B. Putting those together:

exterior=180°−C=A+B\text{exterior} = 180° - C = A + B

The "180°−C180° - C" cancels out the adjacent angle and leaves exactly the two remote ones. In the figure that is 180°−60°=120°180° - 60° = 120°, matching 70°+50°70° + 50°.

Key result

An exterior angle of a triangle equals the sum of the two remote interior angles. (Equivalently, it is 180°180° minus the interior angle right next to it.) Because it adds two positive angles, an exterior angle is always larger than either remote interior angle on its own.

Worked example: A triangle has remote interior angles of 65° and 40° relative to an exterior angle. Find the exterior angle and the interior angle adjacent to it.

  1. The exterior angle equals the sum of the two remote interior angles: 65°+40°=105°65° + 40° = 105°.

    Before the next step: What is the exterior angle?

Practice

Score: 0/4

  1. 1.The two remote interior angles relative to an exterior angle measure 50° and 70°. What is the exterior angle, in degrees?
    °
  2. 2.An exterior angle of a triangle measures 130°. One remote interior angle is 85°. What is the other remote interior angle, in degrees?
    °
  3. 3.An exterior angle measures 5x°, and its two remote interior angles measure (2x + 10)° and (x + 30)°. Find x.
  4. 4.How does an exterior angle of a triangle compare to either of its remote interior angles?
For teachers

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