Define a Right Triangle and Its Parts
If you’re looking at their angles, triangles can be right, acute, or obtuse. So what makes a triangle right? Quite simply, a right triangle has a right angle in it. But it can only have one right angle, because the total number of degrees in a triangle is 180. If it had two right angles, then those two angles would take up all 180 degrees; no degrees would be left for a third angle. So, in a right triangle, the other two angles share the remaining 90 degrees.
Right triangles can come in all sorts of shapes, but they all have that corner, where the right angle sits. In the preceding figure, you see that in all the triangles, the right angle has two sides that are perpendicular to one another. The other two angles are acute angles (meaning they’re less than 90 degrees).
Even though the focus in a right triangle is the right angle, a right triangle actually has six different parts: three angles and three sides. Now, this fact is true of any triangle, but right triangles have special names for these parts. Having special names is necessary because so many properties, theorems, and applications using right triangles are out there, and the names make talking about and explaining the triangles more clear to the reader.
The following figure shows a typical right triangle with standard labels. You see both capital letters and lowercase letters. Since the time of Leonard Euler, the famous Swiss mathematician, this type of labeling has been the tradition. You use lowercase letters to mark the sides of the triangle and capital letters to mark the vertex (angle) opposite each side with the corresponding lowercase letter.
The little square at the vertex C shows that the two sides meeting there at C are perpendicular at that vertex — that’s where the right angle is. The side c, opposite the right angle, is called the hypotenuse. The other two sides, a and b, are called the legs. The hypotenuse is always the longest side, because it’s opposite the largest angle.
In the preceding figure, the angle at vertex C is the right angle, and the other two angles, A and B, are acute angles.
If the angle at vertex A is bigger than that at vertex B, then side a is longer than side b. The measures of a triangle’s angles have a direct relationship to the lengths of the sides opposite them.