Triangles
27. The Acute Triangle · When every angle stays under 90°
All three interior angles remain less than 90°.
An acute triangle has all three interior angles less than 90°. No angle is right or obtuse — every angle is sharp. Drag any vertex; as soon as one angle reaches 90° or more, the triangle is no longer acute.
What this lesson covers
Try to break it
Try pulling one corner far out. See how an angle grows past 90°? Now it's no longer acute. The definition requires ALL angles to stay sharp.
How you build it
Make an acute triangle.
- Place point A — first vertex.
- Place point B not too far from A.
- Place point C so all three angles look smaller than a right angle.
- Draw side AB.
- Draw side BC.
- Draw side CA. All three angles < 90° — that''s an acute triangle.
The proof, step by step
We don't need a long proof for a definition! Just measure each angle with your protractor tool. If every single one reads less than 90°, the triangle earns the 'acute' title.
- Identify the three interior angles of triangle ABC.
- Measure angle A. Observe that it is less than 90°.
- Measure angle B. Observe that it is less than 90°.
- Measure angle C. Observe that it is less than 90°.
- Since all three angles are acute (< 90°), triangle ABC is an acute-angled triangle by definition.
Worked example
In an acute-angled triangle, which of the following statements is always true regarding its interior angles?
By definition, an acute-angled triangle has all three interior angles measuring strictly less than 90°. If any angle reaches or exceeds 90°, it becomes a right or obtuse triangle respectively.
- Exactly one angle is 90°
- One angle is greater than 90°
- All angles are less than 90° — correct
- Two angles are greater than 90°