Question

5 - 5: lesson quiz (honors)
select all the inequalities that relate the sides of the figure shown
a ( b > a )
b ( c > a )
c ( b < c )
d ( a < b < c )
e ( c < b < a )

Explanation:

Step1: Recall the Hinge Theorem

The Hinge 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 side opposite the larger included angle is longer.

Step2: Analyze the angles and sides

In the given figure, for the two triangles with side \(a\) and \(b\):
The included angle for side \(a\) is \(75^{\circ}\), and for side \(b\) is \(90^{\circ}\). Since \(90^{\circ}>75^{\circ}\), by the Hinge Theorem, \(b > a\).
For the two triangles with side \(b\) and \(c\):
The included angle for side \(b\) is \(90^{\circ}\), and for side \(c\) is \(105^{\circ}\). Since \(105^{\circ}>90^{\circ}\), by the Hinge Theorem, \(c > b\).

Answer:

A. \(b > a\), B. \(c > a\), D. \(a < b < c\)