Question

use the diagram
complete the equality or inequality that relates ( mangle b ) to ( mangle d ).
( mangle b ) choose... ( mangle d )

Explanation:

Step1: Apply 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 third side of the first triangle is longer than the third side of the second triangle. Here, in \(\triangle ABC\) and \(\triangle DEF\), \(AC = DE\), \(AB=DF\) (marked as congruent), and \(BC = 12\), \(CD = 10\). Since \(12>10\).

Step2: Compare the angles

By the Hinge Theorem, if \(BC>CD\) (where \(BC\) and \(CD\) are the third - sides of the two triangles \(\triangle ABC\) and \(\triangle DFC\) with two pairs of congruent sides), then \(m\angle BAC>m\angle D\) (the included angles).

Answer:

\(m\angle B>m\angle D\)