Acute-Angled Triangle

An acute-angled triangle is a triangle in which all three interior angles measure less than \(90^\circ\). These angles are called acute angles.

      A
     / \
    /   \
   B-----C
(all angles < 90°)

This type of triangle looks sharp and compact because none of the angles open too wide.

Definition: A triangle is called an acute-angled triangle if each of its three angles is an acute angle, i.e.,

\( \angle A < 90^\circ, \quad \angle B < 90^\circ, \quad \angle C < 90^\circ \)

The triangle may have equal or unequal sides—only the angles matter here.

• All interior angles are less than \(90^\circ\).
• The triangle always forms a compact, narrow shape.
• The orthocentre (point where all altitudes meet) lies inside the triangle.
• Any triangle with no right or obtuse angles is automatically acute-angled.

Here are some quick ways to check if a triangle is acute-angled:

• If all angles look smaller than right angles, the triangle is acute.
• If the triangle appears symmetrical or balanced without any wide opening, it is likely acute.
• If using angle sum, ensure each angle is individually < \(90^\circ\).

Acute triangles appear naturally in many places:

• Some slices of cake or pizza where the tip is very sharp.
• Triangular logos or decorative shapes with narrow angles.
• Support frames or stands designed with narrow triangular supports.
• Triangular road signs that are not equilateral or right-angled.

An acute-angled triangle can be:

• Scalene acute: all sides different but all angles acute.
• Isosceles acute: two equal sides and all angles acute.
• Equilateral: technically also acute, since each angle is \(60^\circ\).

Thus, the acute property depends only on angles, not on side lengths.