A student asks why "congruent" rather than "equal" is used to discuss triangles that have the same size and shape. What do you say?
equal has many meanings, most of them to do with numbers. Maybe equal area, equal perimeter, etc.
congruent is for geometric properties, not numeric values. For instance, we say two angles are congruent when they have equal measures.
To understand why the term "congruent" is used instead of "equal" when discussing triangles, it is important to first understand the difference between the two words in the context of geometry.
In mathematics, the word "equal" is typically used to describe quantities that are exactly the same or identical. For example, if you have two line segments that have the same length, you would say they are equal.
On the other hand, when we talk about shapes, we use the term "congruent" to describe two shapes that are not only equal in size, but also in shape. In other words, congruent figures have the same dimensions and angles, and can be superimposed onto each other by translation, rotation, or reflection.
In the case of triangles, if two triangles have the same size (meaning their corresponding sides and angles are equal), we call them congruent triangles. This means that the triangles not only have equal side lengths, but also equal angles, which ensures that they are identical in shape.
Using the term "congruent" instead of "equal" for triangles emphasizes the fact that they are not just equal in size, but are also identical in shape. While the two words are somewhat similar, "congruent" specifically conveys the idea of shape equality, which is crucial in discussing geometric concepts and properties.