Triangle Angle Sum

Stretch a triangle however you like — tall and thin, wide and flat, a perfect equilateral. Its three corners never agree on their sizes, but they always agree on one thing: their measures add up to $180°$, a straight line's worth of turn.

Explore

Drag a vertex or the side handles to reshape the triangle. Watch the three angle measures change — and watch their sum. Try to make a triangle whose angles don't total 180°.

Make a prediction

Test it

In the tool, push the triangle toward extremes — make one angle nearly $180°$, or squash it almost flat. The individual angles swing wildly, but their sum holds at $180°$ every time. That stubborn constant is the Triangle Angle Sum Theorem.

Why it works

Here is the classic see-it-yourself argument: tear the three corners off a paper triangle and lay them side by side, points touching. They fit together into a perfectly straight line — no gap, no overlap.

Why must they always tile a straight line? Draw a line through the top vertex parallel to the base. The two base angles reappear at the top as alternate interior angles across the parallel lines — and now all three angles of the triangle sit along that one straight line at the top, summing to $180°$. (The tool's "Show why it's 180°" reveal animates this directly: it rotates two of the angles up to the third and lands them on a straight line.)