Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

Verified step by step guidance

1. Identify the measures of the angles in the triangle. If the problem provides side lengths, use the Law of Cosines to find the angles. The Law of Cosines formula is: \(c^2 = a^2 + b^2 - 2ab \cos(C)\), where \(C\) is the angle opposite side \(c\).
2. Classify the triangle by its angles: if all angles are less than 90°, it is an acute triangle; if one angle is exactly 90°, it is a right triangle; if one angle is greater than 90°, it is an obtuse triangle.
3. Classify the triangle by its sides: if all three sides are equal, it is equilateral; if exactly two sides are equal, it is isosceles; if all sides are different lengths, it is scalene.
4. To verify side equality, compare the given side lengths directly. If only angles are given, use the Law of Sines or Law of Cosines to find side lengths if needed.
5. Summarize your classification by combining the angle-based and side-based categories, for example, 'acute isosceles' or 'right scalene'.

Key Concepts

Classification of Triangles by Angles

Triangles are classified based on their angles into acute (all angles less than 90°), right (one angle exactly 90°), or obtuse (one angle greater than 90°). Understanding these categories helps determine the triangle's shape and properties.

Classification of Triangles by Sides

Triangles are also classified by side lengths as equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides different). This classification provides insight into the triangle's symmetry and side relationships.

Relationship Between Angles and Sides in Triangles

The size of a triangle's angles is related to the lengths of its sides: larger sides face larger angles. This principle helps in identifying the triangle type when given side lengths or angles, aiding in accurate classification.