Identify the triangle by its sides and angles.

1. **Classifying by Sides** - If all three sides are equal, i.e., \(a = b = c\), then the triangle is **equilateral**. - If exactly two sides are equal, for instance \(a = b \neq c\), then the triangle is **isosceles**. - If no sides are equal, i.e., \(a \neq b \neq c\), then the triangle is **scalene**. 2. **Classifying by Angles** - Use the fact that the sum of the interior angles is \(180^\circ\): \[ A + B + C = 180^\circ \] - If one of the angles is exactly \(90^\circ\) (for example, \(A = 90^\circ\)), then the triangle is a **right triangle**. - If one of the angles is greater than \(90^\circ\) (say, \(A > 90^\circ\)), then the triangle is **obtuse**. - If all angles are less than \(90^\circ\), then the triangle is **acute**. 3. **Example** Suppose a triangle has sides \(a\), \(b\), and \(c\) and angles \(A\), \(B\), and \(C\): - If \(a = b = c\), then automatically, the triangle is equilateral and each angle is: \[ A = B = C = \frac{180^\circ}{3} = 60^\circ \] Thus, it is also acute. - If \(a = b \neq c\) and one of the angles is \(90^\circ\), then the triangle is both isosceles and right. - If \(a\), \(b\), and \(c\) are all different, and one of the angles exceeds \(90^\circ\), say \(A > 90^\circ\), then the triangle is scalene and obtuse. By comparing the side lengths and measuring the angles, you can identify the classification of the triangle based on the above criteria.