Understanding Triangle Similarity in Rectangles
The image depicts problem number 3, which asks to explain why triangle \(\triangle FE\)C must be similar to triangle \(\triangle ABC\) in the given rectangle \(ABCD\) with the diagonal \(AC\).
From the given diagram, we can observe that rectangle \(ABCD\) consists of \(\triangle ABC\) and \(\triangle ADC\), which are congruent to each other since both are right triangles sharing the diagonal \(AC\) as the hypotenuse.
By the definition of a rectangle, we know:
a) The opposite sides are equal (AD = BC and AB = DC).
b) All angles are right angles (90 degrees).
Point \(F\) lies on \(AB\) and point \(E\) lies on \(DC\), such that \(FE \parallel AC\). Due to the parallel lines, we can deduce the following:
1. Angle \(BAC\) is congruent to angle \(FEC\) because they are corresponding angles.
2. Angle \(ABC\) is congruent to angle \(EFC\) because they are also corresponding angles.
Since \(\triangle ABC\) and \(\triangle FEC\) share the same angle (\(C\)) and have two angles congruent to each other, by the Angle-Angle (AA) postulate, the two triangles are similar.
To summarize, \(\triangle FEC\) is similar to \(\triangle ABC\) because they have two angles congruent by parallel lines (\(FE \parallel AC\)) creating corresponding angles, and they share the angle at \(C\).
