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While working on a problem on finding the shaded region in a rectangle, I realized that if a line cuts a rectangle into two quadrilaterals, then these two quadrilaterals are similar as the opposite sides of a rectangle are parallel. (angle $AFE = $ angle $CEF$). It can be shown that quadrilateral $AFED$ is similar to quadrilateral $CEFB$. If the quadrilaterals are similar, the sides should be in proportion, $$\frac{AF}{CE}=\frac{AD}{CB}$$ and we know that $ad=cb$ from the properties of a rectangle. This should mean that $AF=CE$ but from drawing many rectangles, I understand that this is not the case. Please will someone tell me what I am doing wrong, why $AF$ is not equal to $CE$.

Diagram: enter image description here

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  • $\begingroup$ I know understand that polygons with more than 3 sides may have equal angle but not be similar. thanks @Rory Daulton $\endgroup$
    – axcvbk
    Jul 1, 2015 at 20:43

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The line segment $\overline{EF}$ may be shifted to one side, so that the angles of the two small quadrilaterals are equal but the sides are not proportional.

enter image description here

For triangles, equal angles guarantee similarity (the AA or AAA postulate), but that is not true for polygons with more than three sides. For those you also need to somehow state that the sides are proportional.

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