Inequalities in Two Triangles (Hinge Theorem)

The Hinge Theorem states that if two triangles have two sides of one triangle congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

The angles opposite the equal sides are also equal.

It means that the side opposite the greater angle is longer than the side opposite the smaller angle.

Angle A < Angle D according to the Hinge Theorem.

In a triangle, the larger the angle, the longer the side opposite that angle.

By comparing the lengths of two sides and the included angle of each triangle.

The side opposite angle 1 is longer than the side opposite angle 2.

It means that all corresponding sides and angles of the triangles are equal.

The third sides are equal according to the properties of congruent triangles.

The included angle is crucial because it determines the relationship between the lengths of the opposite sides.