A triangle is a polygon with three sides. Its three sides are line segments, and the three points where the sides join are called the vertices of the triangle (each vertex is also the vertex of one interior angle). A triangle has three interior angles, and the measures of those three angles always add to 180°. Triangles are named by their vertices — for example, triangle ABC — and every triangle is classified in two ways at the same time: once by its angles and once by its sides.
Angles and sides keep the same count. The number of congruent angles in a triangle always matches the number of congruent sides:
| Congruent angles | Congruent sides | Example |
|---|---|---|
| 3 angles congruent | 3 sides congruent | Equiangular and equilateral |
| 2 angles congruent | 2 sides congruent | Isosceles |
| 0 angles congruent | 0 sides congruent | Scalene |
One more naming idea used all through this page: in triangle ABC, the side that does not touch an angle is opposite that angle. Angle A is opposite side , angle B is opposite side , and angle C is opposite side .
What to do: Drag any of the three orange vertex handles. The vertices snap to grid points, and every measure is recomputed from the triangle you have built — the side lengths, the angle measures, and both classifications update together as you drag. Use the example buttons to jump to a known example.
What to notice: the by-angles classification looks only at the angle measures, and the by-sides classification looks only at the side lengths. Matching tick marks on two sides mean those sides are congruent (equal in length), and a small square at a vertex marks a right angle.
Two kinds of triangles have names for their parts. Use the first two buttons to switch between a right triangle and an isosceles triangle, then press the name of a part to highlight it on the diagram and read its definition.
Remember that parts are named by what they are opposite to: the hypotenuse of a right triangle is opposite the right angle, and the base of an isosceles triangle is opposite the vertex angle.
What to do: A target classification is given below. Drag the three vertices (they snap to grid points) until the triangle you have built matches both parts of the target — the by-angles part and the by-sides part. There is more than one triangle that works for every target; any triangle that truly matches both parts counts. Press New Challenge for a new target.
Target:
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.