How is an exterior angle of a triangle related to the interior angles of the triangle? For example, how is exterior angle \begin{align*}\angle BCE\end{align*} related to interior angles \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*}?

### Triangles

Triangles can be classified by their sides and by their angles.

When classifying a triangle by its sides, you should look to see if any of the sides are the same length. If no sides are the same length, then it is a **scalene triangle**. If two sides are the same length, then it is an **isosceles triangle**. If all three sides are the same length, then it is an **equilateral triangle**. You can show that two sides are the same length by drawing tick marks through the middles of the sides. Sides with a corresponding number of tick marks are the same length.

When classifying a triangle by its angles, you should look at the size of the angles. If there is a right angle (a \begin{align*}90^\circ\end{align*} angle), then it is a **right triangle**. If the measures of all angles are less than \begin{align*}90^\circ\end{align*}, then it is an **acute triangle**. If the measure of one angle is greater than \begin{align*}90^\circ\end{align*}, then it is an **obtuse triangle**.

The sum of the measures of the interior angles of any triangle is \begin{align*}180^\circ\end{align*}. If the three angles of a triangle are all the same, then the triangle is an **equiangular triangle** and each angle measure is \begin{align*}60^\circ\end{align*}. Equilateral triangles are always equiangular and vice versa. In fact, the number of sides that are the same length will always correspond to the number of angles that are the same measure.

**Solve the following problems**

Classify the triangle below by its sides and angles.

This triangle has two sides that are congruent (the same length), so it is isosceles. It also has one angle that is greater than \begin{align*}90^\circ\end{align*}, so it is obtuse.

The measures of two angles of a triangle are \begin{align*}30^\circ\end{align*} and \begin{align*}60^\circ\end{align*}. What type of triangle is it?

You know that the measures of the three angles must add up to \begin{align*}180^\circ\end{align*}. This means that the measure of the third angle is \begin{align*}180^\circ-60^\circ-30^\circ=90^\circ\end{align*}, so it is a right triangle. Because all of the angles are different measures, all of the sides must be different lengths, so it is a scalene triangle.

Is it possible for an obtuse triangle to be equilateral?

No. Equilateral triangles are always equiangular and acute. Because the sum of the measures of the angles of a triangle is always \begin{align*}180^\circ\end{align*}, the measures of the three angles in an equiangular/equilateral triangle are each \begin{align*}60^\circ\end{align*}. In order to be obtuse, one of the angles would have to be greater than \begin{align*}90^\circ\end{align*}.

**Examples**

**Example 1**

Earlier, you were asked how is an exterior angle of a triangle related to the interior angles of a triangle.

You know that \begin{align*}m \angle A+m \angle B+m \angle BCA=180^\circ\end{align*}. You also know that \begin{align*}m \angle ECB+m \angle BCA=180^\circ\end{align*} because those two angles form a straight angle and are therefore supplementary.

\begin{align*}& m \angle A+m \angle B+m \angle BCA=180^\circ \\ & m \angle ECB \quad \quad + m \angle BCA=180^\circ\end{align*}

\begin{align*}m \angle A+m \angle B=m \angle ECB\end{align*} angles.*remote interior*. In general, the measure of an exterior angle of a triangle will always be equal to the sum of the measures of the You can see that

#### Example 2

The measures of two angles of a triangle are \begin{align*}45^\circ\end{align*} and \begin{align*}45^\circ\end{align*}. What type of triangle is it?

The third angle must be \begin{align*}90^\circ\end{align*}, so it is a right triangle. Because two angles are the same measure, two sides must be the same length. Therefore, it is isosceles.

#### Example 3

Solve for \begin{align*}x\end{align*} (the picture is not drawn to scale):

Set up an equation and solve:

\begin{align*}x+3x-4+2x+5 &=180 \\ 6x+1 &=180 \\ x &=\frac{179}{6} \approx 29.8\end{align*}

#### Example 4

Find the measure of each angle from \begin{align*}\Delta IJH\end{align*} in #2.

\begin{align*}m \angle J\approx 29.8^\circ\end{align*}; \begin{align*}m \angle I\approx 3(29.8)-4=85.4^\circ\end{align*}; \begin{align*}m \angle H\approx 2(29.8)+5=64.6^\circ\end{align*}.

#### Example 5

Which side of \begin{align*}\Delta IJH\end{align*} is longest? Which side is shortest?

The largest angle is \begin{align*}\angle I\end{align*}, so the longest side must be the side created by \begin{align*}\angle I\end{align*}, which is the side across from \begin{align*}\angle I\end{align*}. The longest side is \begin{align*}\overline{JH}\end{align*}. The shortest side is across from the smallest angle. The smallest angle is \begin{align*}\angle J\end{align*}, so the shortest side is \begin{align*}\overline{IH}\end{align*}.

### Review

1. What are the three ways to classify a triangle by its sides?

2. What are the four ways to classify a triangle by its angles?

3. Can a right triangle be equiangular? Explain.

4. The measures of two angles of a triangle are \begin{align*}42^\circ\end{align*} and \begin{align*}42^\circ\end{align*}. What type of triangle is it?

5. The measures of two angles of a triangle are \begin{align*}120^\circ\end{align*} and \begin{align*}12^\circ\end{align*}. What type of triangle is it?

6. Solve for \begin{align*}x\end{align*} (the picture is not drawn to scale).

7. Find the measure of each angle for \begin{align*}\Delta ABC\end{align*} in #6.

8. Solve for \begin{align*}x\end{align*} (the picture is not drawn to scale).

9. Find the measure of each angle for the triangle in #8.

10. Which side \begin{align*}\Delta DEF\end{align*} of from #8 is the longest? Which side of \begin{align*}\Delta DEF\end{align*} is the shortest? How do you know?

11. Use the angle measurements to order the sides of the triangle below from shortest to longest.

12. Use the side measurements to order the measures of the angles in the triangle below from smallest to largest.

13. One of the exterior angles of a triangle is \begin{align*}100^\circ\end{align*}. What do you know about the interior angles?

14. Solve for \begin{align*}x\end{align*} (the picture is not drawn to scale).

15. Find \begin{align*}m \angle DEC\end{align*} (the picture is not drawn to scale).

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.4.