Question

1. hinge theorem/converse

a. compare the given side lengths
ab ____de
(there is a diagram with triangles abc and def, where angle c is 105°, angle f is 110°, and some sides are marked as congruent)

Explanation:

Step1: Identify 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 smaller than the included angle of the second triangle, then the third side of the first triangle is shorter than the third side of the second triangle.

Step2: Analyze Given Triangles

• In $\triangle ABC$ and $\triangle DEF$:
• $AC \cong DF$ (marked congruent), $BC \cong EF$ (marked congruent).
• Included angle in $\triangle ABC$: $\angle C = 105^\circ$.
• Included angle in $\triangle DEF$: $\angle F = 110^\circ$.

Step3: Apply Hinge Theorem

Since $105^\circ < 110^\circ$ (i.e., $\angle C < \angle F$), by the Hinge Theorem, the side opposite the smaller angle ($AB$) is shorter than the side opposite the larger angle ($DE$). So, $AB < DE$.

Answer:

$AB < DE$