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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`

`Thus`

`HF^2=HE^2+EF^2```

`HF^2=51^2+28^2` , by Pythagoras Theorem.

Also

in `DeltaEHG`

`HE=28,HG=51=EF`

so

`HF^2=HE^2+HG^2=EG^2`

`HF=EG`

Thus triangle EHG congruent to triangle HEF by (SSS) crietrion.

this implies

`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 ).

HE=EH

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