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 types, for example, 'acute isosceles' or 'right scalene'.

Key Concepts

Classification of Triangles by Angles

Triangles are classified based on their angles as 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: equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides different). This classification aids in identifying symmetry and congruence within the triangle.

Relationship Between Triangle Angles and Sides

The type of angles in a triangle influences the relative lengths of its sides, and vice versa. For example, an equilateral triangle is always acute, while an obtuse triangle cannot be equilateral. Recognizing these relationships is key to accurate classification.