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# Triangle Classification

## Categories of triangles based on angle measurements or the number of congruent sides.

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Triangles

How is an exterior angle of a triangle related to the interior angles of the triangle? For example, how is exterior angle BCE\begin{align*}\angle BCE\end{align*} related to interior angles A\begin{align*}\angle A\end{align*} and B\begin{align*}\angle B\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 90\begin{align*}90^\circ\end{align*} angle), then it is a right triangle. If the measures of all angles are less than 90\begin{align*}90^\circ\end{align*}, then it is an acute triangle. If the measure of one angle is greater than 90\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 180\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 60\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.

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 90\begin{align*}90^\circ\end{align*}, so it is obtuse.

Example B

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

Solution: You know that the measures of the three angles must add up to 180\begin{align*}180^\circ\end{align*}. This means that the measure of the third angle is 1806030=90\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.

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 180\begin{align*}180^\circ\end{align*}, the measures of the three angles in an equiangular/equilateral triangle are each 60\begin{align*}60^\circ\end{align*}. In order to be obtuse, one of the angles would have to be greater than 90\begin{align*}90^\circ\end{align*}.

Concept Problem Revisited

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

mA+mB+mBCA=180mECB+mBCA=180

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

#### Vocabulary

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.

right triangle has one right angle.

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

An acute triangle has all three angles with measures less than 90\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 45\begin{align*}45^\circ\end{align*} and 45\begin{align*}45^\circ\end{align*}. What type of triangle is it?

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

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

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

1. The third angle must be 90\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.

2. Set up an equation and solve:

x+3x4+2x+56x+1x=180=180=179629.8

3. mJ29.8\begin{align*}m \angle J\approx 29.8^\circ\end{align*}; mI3(29.8)4=85.4\begin{align*}m \angle I\approx 3(29.8)-4=85.4^\circ\end{align*}; mH2(29.8)+5=64.6\begin{align*}m \angle H\approx 2(29.8)+5=64.6^\circ\end{align*}.

4. The largest angle is I\begin{align*}\angle I\end{align*}, so the longest side must be the side created by I\begin{align*}\angle I\end{align*}, which is the side across from I\begin{align*}\angle I\end{align*}. The longest side is JH¯¯¯¯¯\begin{align*}\overline{JH}\end{align*}. The shortest side is across from the smallest angle. The smallest angle is J\begin{align*}\angle J\end{align*}, so the shortest side is IH¯¯¯¯¯\begin{align*}\overline{IH}\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 42\begin{align*}42^\circ\end{align*} and 42\begin{align*}42^\circ\end{align*}. What type of triangle is it?

5. The measures of two angles of a triangle are 120\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).

### Vocabulary Language: English

Acute Triangle

Acute Triangle

An acute triangle has three angles that each measure less than 90 degrees.
Interior angles

Interior angles

Interior angles are the angles inside a figure.
Isosceles Triangle

Isosceles Triangle

An isosceles triangle is a triangle in which exactly two sides are the same length.
Obtuse Triangle

Obtuse Triangle

An obtuse triangle is a triangle with one angle that is greater than 90 degrees.
Right Angle

Right Angle

A right angle is an angle equal to 90 degrees.
Scalene Triangle

Scalene Triangle

A scalene triangle is a triangle in which all three sides are different lengths.
Equilateral

Equilateral

A polygon is equilateral if all of its sides are the same length.
Equiangular

Equiangular

A polygon is equiangular if all angles are the same measure.