Inequalities in Two Triangles

Hinge Theorem (SAS Inequality Theorem)

If two sides of one Triangle are congruent to two sides of another Triangle and the included Angle of the first is larger than the included Angle of the second, then the third side of the first Triangle is longer than the third side of the second Triangle.

Converse of the Hinge Theorem (SSS Inequality)

If two sides of one Triangle are congruent to two sides of another Triangle and the third side of the first is longer than the third side of the second, then the included Angle of the first Triangle is larger than the included Angle of the second.

Analogy

Think of a door hinge: two sides of the Triangle are like the door and the frame, and the included Angle is how wide the door is open. A wider opening Angle means the gap (third side) is larger.

Example

In △ABC and △DEF: AB = DE = 8, BC = EF = 6, and ∠B = 70° while ∠E = 50°.

Since ∠B > ∠E and the two pairs of sides are equal, the Hinge Theorem tells us AC > DF.