Question

in the triangles, (overline{bc} cong overline{de}) and (overline{ac} cong overline{fe}). if (mangle c) is greater than (mangle e), then (overline{ab}) is ______ (overline{df}). (\bigcirc) congruent to (\bigcirc) longer than (\bigcirc) shorter than (\bigcirc) the same length as (with an image of two triangles: triangle (abc) with vertices (a), (c), (b) and triangle (dfe) with vertices (f), (e), (d), showing congruent side markings on (overline{ac} cong overline{fe}) and (overline{bc} cong overline{de}))

Explanation:

Step1: Recall the Hinge Theorem

The Hinge Theorem (also known as the 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 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: Identify the triangles and sides/angle

We have two triangles: $\triangle ABC$ and $\triangle DFE$.

• $\overline{BC} \cong \overline{DE}$ (given)
• $\overline{AC} \cong \overline{FE}$ (given)
• The included angle for $\overline{BC}$ and $\overline{AC}$ in $\triangle ABC$ is $\angle C$, and the included angle for $\overline{DE}$ and $\overline{FE}$ in $\triangle DFE$ is $\angle E$.
• We know that $m\angle C > m\angle E$.

Step3: Apply the Hinge Theorem

By the Hinge Theorem, since two sides of $\triangle ABC$ ($\overline{BC}$ and $\overline{AC}$) are congruent to two sides of $\triangle DFE$ ($\overline{DE}$ and $\overline{FE}$) and the included angle $\angle C$ is larger than the included angle $\angle E$, the third side of $\triangle ABC$ (which is $\overline{AB}$) must be longer than the third side of $\triangle DFE$ (which is $\overline{DF}$).

Answer:

longer than