Final answer:
A triangle with no congruent sides is a scalene triangle, while a triangle with two equal sides is an isosceles triangle. These distinctions matter in mathematics because they define properties and solutions applicable in geometry and trigonometry.
Step-by-step explanation:
A triangle with no congruent sides is called a scalene triangle. This means that all three sides of the triangle are of different lengths. Conversely, a triangle with two sides equal is known as an isosceles triangle. This type of triangle has two sides that are the same length and one side that is different.
In mathematics, it is crucial to make these distinctions because each type of triangle has unique properties that affect how it is studied and applied. For instance, specific formulas like the Law of Sines or the Law of Cosines may be more appropriate for finding missing sides or angles in these triangles. Understanding the different types also aids in solving geometric problems and proofs.
Trigonometry focuses on the relationships between triangles' angles and sides, and these relationships are key to many applications in mathematics and the sciences. It's like understanding the fundamental building blocks of geometry since a triangle's shape and side lengths influence its angles, and vice versa; to think of a triangle, we automatically consider a three-sided figure on a plane with angles adding up to 180 degrees.