Classifying Triangles

Geometry · Axiom Academy

Every triangle has two names at once — one from the lengths of its sides, one from the size of its angles. 1. Two Ways to Read a Triangle Start with the sides . Watch the same triangle reshape itself: as the matching tick marks appear and vanish, it moves through the three side-families — all three sides equal, then exactly two equal, then none equal. Sorting by side length gives exactly three families. The tick marks on each figure show which sides are equal. All three sides are equal in length. Two sides are equal in length. Now ignore the sides and look at the angles . Watch the apex slide so the largest angle grows: first all three angles are sharp, then one reaches exactly 90°, then one opens past 90°. All three angles are less than 90°. One angle is greater than 90°. The two systems are independent , so every triangle gets one name from each — a side name and an angle name. The animation reads off both labels for the same shape; for instance, two equal sides with a 90° corner is an isosceles right triangle. Almost every side/angle pairing is possible. The one constraint to remember: an equilateral triangle's angles are each exactly 60°, so it is always acute — it can never be right or obtuse. You can now name any triangle twice over — once by its sides, once by its angles — and combine the two into a single precise label. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Geometry course.