Question

in the triangles, $overline{ac} cong overline{fe}$ and $overline{bc} cong overline{de}$. which statement correctly compares $overline{ab}$ and $overline{fd}$?
options:

• $overline{ab}$ and $overline{fd}$ are the same length.
• $overline{ab}$ is shorter than $overline{fd}$.
• $overline{ab}$ is longer than $overline{fd}$.
• $overline{ab}$ is shorter than or the same length as $overline{fd}$.

(with two triangles: triangle abc with angle at c is 72°, $overline{ac}$ and $overline{bc}$ have markings; triangle fed with angle at e is 65°, $overline{fe}$ and $overline{de}$ have markings)

Explanation:

Step1: Analyze Triangle Properties

In $\triangle ABC$ and $\triangle FDE$, we know $\overline{AC} \cong \overline{FE}$ (given, marked with two ticks) and $\overline{BC} \cong \overline{DE}$ (given, marked with one tick). So, these are two sides of the triangles. Now, we look at the included angles: $\angle C = 72^\circ$ and $\angle E = 65^\circ$.

Step2: Apply 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.

Here, $\overline{AC} \cong \overline{FE}$, $\overline{BC} \cong \overline{DE}$, and $\angle C = 72^\circ > \angle E = 65^\circ$. So, by the Hinge Theorem, the third side opposite the larger angle (which is $\overline{AB}$ in $\triangle ABC$ and $\overline{FD}$ in $\triangle FDE$) will have $\overline{AB} > \overline{FD}$.

Answer:

$\overline{AB}$ is longer than $\overline{FD}$