Math Core

Draw any triangle you like: tall and skinny, short and wide, or perfectly balanced. Measure its three angles and add them up, and you will always get the same total. In this lesson you'll see why, using the parallel-line facts from the last lesson.

The angles of a triangle add to 180°

Start with triangle $ABC$ , with angles $a$ , $b$ and $c$ . Draw a line through $C$ that is parallel to side $AB$ .

The line through C is parallel to AB. The angles along it at C match the triangle's angles.

Now look at the three angles that sit along the new line at $C$ :

The angle on the left and angle $a$ are alternate interior angles (side $AC$ is the transversal), so they are equal.
The angle on the right and angle $b$ are also alternate interior angles (side $BC$ is the transversal), so they are equal.
The middle angle is angle $c$ itself.

Those three angles together make a straight line, which is $180^\circ$ . So $a + b + c = 180^\circ$ .

Triangle angle sum

The three interior angles of every triangle add up to $180^\circ$ .

m\angle A + m\angle B + m\angle C = 180^\circ

Worked example: Finding the third angle

Two angles of a triangle measure $47^\circ$ and $68^\circ$ . Find the third angle.

180 - 47 - 68 = 65

The third angle is $65^\circ$ .

Worked example: Using algebra

The angles of a triangle measure $x^\circ$ , $2x^\circ$ and $3x^\circ$ . Find each angle.

\begin{aligned} x + 2x + 3x &= 180 \\ 6x &= 180 \\ x &= 30 \end{aligned}

The angles are $30^\circ$ , $60^\circ$ and $90^\circ$ . It's a right triangle.

Exterior angles

If you extend one side of a triangle past a vertex, the angle between the extension and the other side is an exterior angle. The two interior angles that are not next to it are called the remote interior angles.

Angle 4 is an exterior angle. Angles 1 and 2 are its remote interior angles.

Angles 3 and 4 form a linear pair, so $\angle 3 + \angle 4 = 180^\circ$ . The triangle angle sum says $\angle 1 + \angle 2 + \angle 3 = 180^\circ$ . Both sums equal $180^\circ$ and both include $\angle 3$ , so $\angle 4 = \angle 1 + \angle 2$ .

Exterior angle theorem

An exterior angle of a triangle equals the sum of the two remote interior angles.

Worked example: An exterior angle with algebra

An exterior angle of a triangle measures $(3x - 10)^\circ$ . Its remote interior angles measure $(x + 30)^\circ$ and $50^\circ$ . Find the exterior angle.

\begin{aligned} 3x - 10 &= (x + 30) + 50 \\ 3x - 10 &= x + 80 \\ 2x &= 90 \\ x &= 45 \end{aligned}

The exterior angle is $3(45) - 10 = 125^\circ$ . Check: the remote interior angles are $75^\circ$ and $50^\circ$ , and $75 + 50 = 125$ . ✓

Common mistake

An exterior angle equals the sum of the two remote interior angles. It is supplementary only to the interior angle right next to it. Don't subtract the remote angles from $180^\circ$ .

Angle-angle similarity

Here is a useful consequence. If two angles of one triangle match two angles of another, the third angles must match too, because each third angle is $180^\circ$ minus the same two numbers. Triangles with all three angles equal have the same shape, so they are similar.

Angle-angle (AA) similarity

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Worked example: Checking for similarity

$\triangle ABC$ has angles of $40^\circ$ and $75^\circ$ . $\triangle DEF$ has angles of $40^\circ$ and $65^\circ$ . Are the triangles similar?

The third angle of $\triangle ABC$ is $180 - 40 - 75 = 65^\circ$ . So $\triangle ABC$ has angles $40^\circ$ , $75^\circ$ and $65^\circ$ . Two of them ( $40^\circ$ and $65^\circ$ ) match angles of $\triangle DEF$ , so the triangles are similar.

Tip

After you find a missing angle, add all three angles to check that the total is exactly $180^\circ$ .