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*}

#### Watch This

Watch the second part of this video:

http://www.youtube.com/watch?v=z_O2Knid2XA James Sousa: Types of Triangles

#### Guidance

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*}**right triangle**. If the measures of all angles are less than \begin{align*}90^\circ\end{align*}**acute triangle**. If the measure of one angle is greater than \begin{align*}90^\circ\end{align*}**obtuse triangle**.

The sum of the measures of the interior angles of any triangle is \begin{align*}180^\circ\end{align*}**equiangular triangle** and each angle measure is \begin{align*}60^\circ\end{align*}

**Example A**

Classify the triangle below by its sides and angles.

**Solution:** 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*}

**Example B**

The measures of two angles of a triangle are \begin{align*}30^\circ\end{align*}

**Solution:** You know that the measures of the three angles must add up to \begin{align*}180^\circ\end{align*}

**Example C**

Is it possible for an obtuse triangle to be equilateral?

**Solution:** 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*}

**Concept Problem Revisited**

You know that \begin{align*}m \angle A+m \angle B+m \angle BCA=180^\circ\end{align*}

\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*}

You can see that \begin{align*}m \angle A+m \angle B=m \angle ECB\end{align*}*remote interior* angles.

#### Vocabulary

A ** scalene triangle** has no sides that are congruent.

An ** isosceles triangle** has two sides that are congruent.

An ** equilateral triangle** has three sides that are congruent.

A ** right triangle** has one right angle.

An ** obtuse triangle** has one angle with a measure that is greater than \begin{align*}90^\circ\end{align*}

An ** acute triangle** has all three angles with measures less than \begin{align*}90^\circ\end{align*}

An ** equiangular triangle** has three congruent angles.

#### Guided Practice

1. The measures of two angles of a triangle are \begin{align*}45^\circ\end{align*}

2. Solve for \begin{align*}x\end{align*}

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

4. Which side of \begin{align*}\Delta IJH\end{align*}

**Answers**:

1. The third angle must be \begin{align*}90^\circ\end{align*}

2. 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*}

3. \begin{align*}m \angle J\approx 29.8^\circ\end{align*}

4. The largest angle is \begin{align*}\angle I\end{align*}

#### Practice

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*}

5. The measures of two angles of a triangle are \begin{align*}120^\circ\end{align*}

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).