1. The mathematician was studying congruences to prove a theorem.
数学家正在研究同余以证明一个定理。

2. The two triangles are congruent because their corresponding sides are equal in length.
这两个三角形是全等的，因为它们对应的边长相等。

3. The congruences between the two geometric shapes were evident to the students.
这两个几何图形之间的全等关系对学生们来说是显而易见的。

4. The teacher explained how to use congruences to find missing angles in a triangle.
老师解释了如何利用同余来找到三角形中缺失的角度。

5. The student struggled to understand the concept of congruences in geometry class.
学生在几何课上很难理解同余的概念。

6. The theorem about congruences in modular arithmetic was particularly difficult for the class to grasp.
关于模算术中的同余定理对班上的学生来说尤为困难。