Obtuse-Angled Triangle

An obtuse-angled triangle is a triangle in which one of the interior angles is greater than \(90^\circ\). This wide opening makes the triangle appear stretched or spread out.

      A
     / \
    /   \
   B-----C
  (one angle > 90°)

The other two angles must always be acute angles (less than \(90^\circ\)).

Definition: A triangle is called an obtuse-angled triangle if exactly one of its three angles is an obtuse angle—that is, an angle measuring more than \(90^\circ\).

Symbolically, if \( \angle A \) is obtuse, then:

\( \angle A > 90^\circ \)

Only one angle can be obtuse, because the total angle sum cannot exceed \(180^\circ\).

• One interior angle is greater than \(90^\circ\).
• The other two angles are always acute (less than \(90^\circ\)).
• The side opposite the obtuse angle is the longest side of the triangle.
• The orthocentre (intersection of altitudes) lies outside the triangle.
• It can be scalene or isosceles, but never equilateral or right-angled.

You can identify an obtuse triangle by checking the angle that looks the widest or most spread-out. That angle is usually the obtuse angle.

• If the largest angle > \(90^\circ\), it is an obtuse-angled triangle.
• The side opposite this angle is the longest side.
• The vertex of the obtuse angle appears to 'open out' compared to others.

Obtuse triangles appear in many natural and constructed shapes:

• Some roof designs with one wide interior angle.
• A folded piece of paper with a wide crease angle.
• Irregular triangular garden or land plots.
• Art designs with stretched triangular shapes.

An obtuse-angled triangle can be:

• Scalene obtuse: all sides unequal, one angle obtuse.
• Isosceles obtuse: two equal sides, one obtuse angle.

It can never be equilateral or right-angled, because those require specific angle conditions.