Our study of triangles begins with their different classifications. But before we can do this, we must learn how to name triangles. Since triangles are defined by their three vertices, we use the triangle symbol, ?, followed by the three vertices (in any order). For instance, ?ABC describes a triangle whose vertices are the points A, B, and C. We can place the points in any order and still describe the same triangle.
This triangle can also be called ?BCA, ?CAB, ?ACB, ?CBA, or ?BAC.
Now that we understand the notation for triangles, we can begin classifying them. There are two ways by which we can classify triangles. One way is by determining the measures of a triangle's angles. Another way in which triangles are classified is by the lengths of their sides. We will utilize both types of triangle classifications to aid in proofs throughout this section.
Classifying Triangles by Angles
A triangle whose three angles are acute is called an acute triangle. That is, if all three angles of a triangle are less than 90°, then it is an acute triangle.
Every angle in these triangles is acute.
An obtuse triangle is a triangle that has one obtuse angle.
The obtuse angles in the triangles above are at vertex H and K, respectively.
A triangle that has one angle that is a right angle is called a right triangle. In other words, if one angle of a triangle is 90°, then it is a right triangle.
If all three angles of a triangle are congruent, then the triangle is an equiangular triangle. Later on, we will learn why the only angle measure possible for equiangular triangles is 60°.
Classifying Triangles by Sides
A triangle with three congruent sides is called an equilateral triangle.
The tick marks indicate congruence between all three sides.
If a triangle has at least two congruent sides, then the triangle is an isosceles triangle. Note that, by definition, equilateral triangles can also be classified as isosceles.
A triangle that has no congruent sides is called a scalene triangle.
No two sides of the triangle above are congruent.
(1) Classify the triangle below as acute, obtuse, right, or equiangular.
Solution: If we look at ?W and ?V, we notice that both angles are acute angles. While this makes us lean toward calling it an acute triangle, we have to check the third angle. Since ?U has a measure of 90°, we know that ?UVW the triangle is actually a right triangle. Had ?U been any less than 90°, the triangle would have been an acute triangle.
(2) Determine the lengths of the sides of the equilateral triangle below.
Solution: Given the fact that the triangle is equilateral, we can set any pair of sides of the triangle equal to each other. In this case, we will show that the length of side AB is equal to the length of side BC in order to solve for x.
Now that we've determined the value of x, we can plug this value into any of the sides of the triangle. We plug it into the equation for side AB below.
We can choose to generalize and say that the other sides of the triangle are also 24 units in length (since it is an equilateral triangle). However, we choose to check our answer to make sure of this. Thus we plug into the equation for side BC first.
Indeed, BC is also 24 units long. Finally, we can plug x = 4 into the equation for CA to assure ourselves that we are correct.