Question

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

In two triangles, 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: Analyze the triangles with sides \(a\) and \(b\)

The two triangles with sides \(a\) and \(b\) have two pairs of congruent sides (the marked equal - length segments). The included angle for side \(a\) is \(75^{\circ}\), and the included angle for side \(b\) is \(90^{\circ}\). Since \(90^{\circ}>75^{\circ}\), by the Hinge Theorem, \(b > a\).

Step3: Analyze the triangles with sides \(b\) and \(c\)

The two triangles with sides \(b\) and \(c\) have two pairs of congruent sides (the marked equal - length segments). The included angle for side \(b\) is \(90^{\circ}\), and the included angle for side \(c\) is \(105^{\circ}\). Since \(105^{\circ}>90^{\circ}\), by the Hinge Theorem, \(c > b\).

Answer:

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