5.2.2 Inequalities in Two Triangles (continued) Example 2 Construction Site Application

In this example, the Hinge Theorem is applied to a real-world case. In the example the boom of the same crane rests at two different positions, so the length of the boom and the height of the tower in each case are the same. These are the lengths of two of the sides of the two triangles formed by the crane. The third sides of the triangles are the distance between the worker at the end of the boom and the base of the tower. For crane B the measure of the included angle made by the boom and the tower is greater than the measure of the included angle made by crane A. Therefore, by the Hinge Theorem, the third side, or the distance between the worker and the crane base, is longer in crane B than crane A. Worker B is farther from the base than worker A.

Example 3 Proving Triangle Relationships

The conjecture to prove in this example is that the length of one side of one triangle is greater than the length of one side of another triangle. The given information is that AB = AC . Begin by making a plan for the proof. Then use the plan to set out the specific steps of the proof and their justifications. Statement 1 is the given information. Statement 2 is true because the two lines are the same line. Notice that two sides of △ ABD are congruent to two sides of △ ACD . Now consider the included angle. According to the Segment Addition Postulate m ∠ DAB = m ∠ DAC + m ∠ CAB . This means, by the Comparison Property of Inequality, that m ∠ DAB > m ∠ DAC . Since m ∠ DAB is greater than m ∠ DAC , the length of the side opposite ∠ DAC , BD , must be greater than the length of the side opposite ∠ DAC , CD , thus proving the conjecture.

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