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A carpenter must ensure that a large window frame is a rectangle. If the corners are labled E, F, G, and H the carpenter knows that EF=51 inches, FG=28 inches, GH=51 inches and EH=28 inches. The carpenter could measure the diagonals EG and FH and verify that they are the same length, but chooses another approach. After verifying that angle E is a right angle the carpenter insists that EFGH must be a true rectangle.
EFGH is a parallelogram since____________?
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It's enough to veirfy that a diagonal D:
`D=sqrt(EF^2+FG^2)` for it'st true if only if the angle are of 90°
Indeed, if the angle `phi != 90°` the diagonal, according Carnot's theorem, is:
`D=sqrt(EF^2+FG^2-2EF xx FG cos phi)`
This equation is equal of that above is only if `cos phi=0` that is `phi=90°`
In ` DeltaHEF , angleE=90^o`
`HF^2=51^2+28^2` , by Pythagoras Theorem.
Thus triangle EHG congruent to triangle HEF by (SSS) crietrion.
`angleEHG=angleHEF=90` ,But these are co-interior angles
Thus cointeror angles are supplementry ,thus
HG is parallel to EF.
Therefore EFGH are parallelogram ( opposite sides are parallel and equal 51 inch ).
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