Question

1. what could you prove about the following diagram, using the hinge theorem?

Explanation:

Step1: Recall Hinge Theorem

The Hinge Theorem (SAS Inequality Theorem) states that if two sides of one triangle are congruent to two sides of another triangle, but 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.

Step2: Identify Congruent Sides and Angles

In the diagram, \( HI = HK \) (marked with congruency ticks) and \( HJ \) is common to both triangles \( \triangle HIJ \) and \( \triangle HKJ \). The included angles are \( \angle IHJ = 34^\circ \) and \( \angle KHJ = 31^\circ \), so \( \angle IHJ > \angle KHJ \).

Step3: Apply Hinge Theorem

By the Hinge Theorem, since two sides (\( HI = HK \), \( HJ = HJ \)) are congruent and the included angle \( \angle IHJ \) is larger than \( \angle KHJ \), the third side \( IJ \) must be longer than \( KJ \), i.e., \( IJ > KJ \).

Answer:

Using the Hinge Theorem, we can prove that \( IJ > KJ \) (or \( KJ < IJ \)).