Question - Understanding Triangle Similarity in Rectangles
Solution:
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 $$C$$.
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