How does Book I of the Elements use triangles and parallelograms?

In Book I of the Elements, Euclid establishes the foundational principles of plane geometry by focusing on the properties and relationships of triangles and parallelograms.12

Key aspects of Book I include:

• Congruence: Euclid develops the theory of congruence for triangles, establishing conditions under which two triangles are identical in shape and size. This includes the famous proof regarding isosceles triangles, historically known as the Bridge of Asses.13
• Parallel Lines: The book introduces the parallel postulate, which is essential for proving theorems such as the sum of a triangle's interior angles being equal to two right angles (180 degrees).13
• Area Relations: Euclid explores the area relationships between these figures, demonstrating that the area of a triangle is half that of a parallelogram with the same base and height. He also proves that triangles sharing the same base and lying between the same parallel lines have equal areas.14

These investigations culminate in the Pythagorean theorem, which Euclid demonstrates using a geometric construction often called the "Windmill proof," relying on the previously established properties of triangles and parallelograms.24