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Geometry Triangle theorems

Triangle angle sum and exterior angle theorems

20 practice questions 2 video lessons Theory + worked examples

Triangle Angle Sum and Exterior Angle

Common Core Geometry • Standard G-CO.10 • Triangle Theorems

Triangle Angle Sum and Exterior Angle is the opening topic of Triangle Theorems in the Common Core State Standards. It is aligned to Standard G-CO.10, which requires students to prove theorems about triangles, including that the measures of interior angles of a triangle sum to 180 degrees.

The interior angles of a triangle sum to \(180^\circ\), and an exterior angle equals the sum of the two remote interior angles.

Common Core Geometry › Triangle Theorems › Triangle Angle Sum and Exterior Angle  —  Standard G-CO.10

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Practice questions

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Watch 2 video(s)
  • Triangle Angle Sum Theorem (and Exterior Angle Theorem) Watch
  • Exterior Angle Theorem For Triangles, Practice Problems - Geometry Watch
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Theory

Two theorems govern the angles of every triangle:

  • Triangle Angle Sum: the three interior angles add to \(180^\circ\).
  • Exterior Angle Theorem: an exterior angle equals the sum of the two remote (non-adjacent) interior angles.

An exterior angle is formed by extending one side of the triangle; it is supplementary to the interior angle next to it.

The exterior angle theorem follows from the angle sum: the exterior angle and its adjacent interior angle make \(180^\circ\), and so do the three interior angles.
Triangle angle sum The three interior angles of a triangle add to 180 degrees. a b c a + b + c = 180°
The interior angles sum to \(180^\circ\).
Exterior angle theorem An exterior angle of a triangle equals the sum of the two remote interior angles. x y ext exterior = x + y (remote interiors)
An exterior angle equals the sum of the two remote interiors.

The two theorems:

\[a+b+c=180^\circ,\qquad \text{exterior}= \text{remote}_1+\text{remote}_2\]
interior angles sum to 180; an exterior angle equals the sum of the two remote interior angles
An exterior angle is always larger than either remote interior angle it equals the sum of.

How to find a triangle angle

  1. Angle sum: subtract the two known angles from \(180^\circ\).
  2. Exterior angle: add the two remote interiors, or subtract to find a missing remote interior.
  3. Algebra: set the sum of angle expressions equal to \(180^\circ\) and solve.
Example 1 — Find the third angle
Two angles of a triangle are \(70^\circ\) and \(50^\circ\). Find the third.
Solution

The three angles sum to \(180^\circ\).

\(180^\circ-70^\circ-50^\circ\)\(=\)\(60^\circ\)
the third angle is 60 degrees
Example 2 — Exterior angle
An exterior angle of a triangle has remote interior angles \(40^\circ\) and \(75^\circ\). Find it.
Solution

The exterior angle equals the sum of the two remote interior angles.

\(40^\circ+75^\circ\)\(=\)\(115^\circ\)
the exterior angle is 115 degrees
Example 3 — Solve with algebra
The angles of a triangle are \(x\), \(2x\), and \(3x\). Find \(x\).
Solution

Set their sum equal to \(180^\circ\).

\(x+2x+3x\)\(=\)\(180\)
\(6x\)\(=\)\(180\)
\(x\)\(=\)\(30\)

The angles are \(30^\circ,60^\circ,90^\circ\).

x equals 30, giving angles 30, 60, and 90
Example 4 — Remote interior from an exterior angle
An exterior angle is \(120^\circ\); one remote interior angle is \(45^\circ\). Find the other remote interior angle.
Solution

Exterior \(=\) sum of remote interiors, so subtract.

\(120^\circ-45^\circ\)\(=\)\(75^\circ\)
the other remote interior angle is 75 degrees

Common pitfalls

The exterior angle uses the remote interiors, not the adjacent one.
All three interior angles sum to \(180^\circ\), not \(360^\circ\). That is quadrilaterals.
An exterior angle is supplementary to its adjacent interior angle — a second way to find it.

Frequently asked questions

What do the angles of a triangle add up to?

Always \(180^\circ\), for every triangle.

What is the exterior angle theorem?

An exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles.

What is a remote interior angle?

An interior angle not adjacent to a given exterior angle — the two that the exterior angle equals the sum of.

How do you find a missing angle in a triangle?

Subtract the two known interior angles from \(180^\circ\), or use the exterior angle theorem.

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