7. Triangles: Classifying by Angles

Triangles can also be classified by their largest angle.

If (L) is the largest angle, then
\[ L<90^\circ\Rightarrow\text{acute},\qquad L=90^\circ\Rightarrow\text{right},\qquad L>90^\circ\Rightarrow\text{obtuse}. \]

• Acute triangle: all three angles are less than \(90^\circ\).
• Right triangle: one angle is exactly \(90^\circ\).
• Obtuse triangle: one angle is greater than \(90^\circ\).

A triangle can have at most one right or obtuse angle, because all three angles must add to \(180^\circ\) (the next lesson). Two right angles alone would already use up \(180^\circ\), leaving nothing for the third.

Worked example: a triangle with angles \(90^\circ\), \(60^\circ\), and \(30^\circ\) is a right triangle. A triangle with angles \(100^\circ\), \(50^\circ\), and \(30^\circ\) is obtuse.

Case study - the corner of a set square (drafting tool) is a right triangle, used to draw perpendicular lines.

Common mistake: thinking a triangle could have two right angles. It cannot -- the angle sum is fixed at \(180^\circ\), so only one angle can be \(90^\circ\) or more.

Use the largest angle

After confirming that the angles total (180^°), inspect the largest angle. If the largest is less than (90^°), every angle is acute. If it equals (90^°), the triangle is right. If it exceeds (90^°), the triangle is obtuse.

A triangle has two names when classified in both ways. For example, side lengths (5,5,8) make it isosceles by sides. If its angles are approximately (37^°,37^°,106^°), it is obtuse by angles. Its full description is an obtuse isosceles triangle.

Reasonableness check: angles (70^°,60^°,50^°) form an acute triangle. Angles (90^°,60^°,40^°) cannot form any triangle because their sum is (190^°).