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COR. Two rectangles are equal, if two adjoining sides of the one are respectively equal to two adjoining sides of the other; and two squares are equal, if a side of the one is equal to a side of the other.

THEOR. 31. If a quadrilateral has two opposite sides equal and parallel, it is a parallelogram.

Let the quadrilateral ABCD have the opposite sides AD and BC equal and parallel :

B

then shall ABCD be a parallelogram.

Join AC.

Then because AC meets the parallel straight lines AD, BC, therefore the angle CAD is equal to the alternate angle ACB.

Hence, in the triangles CAD, ACB,

the side AD is equal to the side CB,

the side AC is common to both,

I. 23.

Нур.

and the angle CAD is equal to the angle ACB,

therefore the angle ACD is equal to the angle CAB,

I. 5.

and these are alternate angles which AC makes with AB and

DC,

therefore AB is parallel to DC,

therefore ABCD is a parallelogram.

I. 21.

Q.E.D.

DEF. 40. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted between perpendiculars let fall on it from the extremities of the former.

THEOR. 32. Straight lines that are equal and parallel have equal projections on any other straight line; conversely, parallel straight lines that have equal projections on another straight line are equal.

Let AB, CD be two parallel straight lines, GH, KL their projections on any other straight line EF:

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then AGHB and CKLD are parallelograms,
therefore AB is equal to its projection GH,
and CD is equal to its projection KL,
therefore, if AB is equal to CD,
then GH is equal to KL;

and conversely, if GH is equal to KL,
then AB is equal to CD.

I. 29.

1. 29.

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draw GM parallel to AB, and KN to CD,

meeting BH and DL produced if necessary, at M and N respectively.

CD,

Then because GM is parallel to AB, KN to CD, and AB to

therefore GM is parallel to KN,

I. 24.

I. 23, Cor. 2.

therefore the angle MGH is equal to the angle NKL.

Hence in the triangles MGH, NKL,

the angle MGH is equal to the angle NKL,

and the angle GHM is equal to the angle KLN, each being a

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THEOR. 33. Equal straight lines that have equal projections on another straight line are parallel to that line, or make equal angles with it.

Let AB, CD be two equal straight lines, GH, KL their

projections on another straight line EF, and let GH be equal to KL:

then shall AB and CD be parallel to EF, or make equal angles with EF.

EF,

If either AB or CD is parallel to EF, let AB be parallel to

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then AGHB is a parallelogram,

and therefore AB is equal to its projection GH, therefore CD is also equal to its projection KL;

but the perpendicular from C upon DL is equal to KL,

therefore CD is this perpendicular,

and therefore CD is parallel to EF.

I. 29. Hyp.

I. 29.

I. 15.

Therefore AB and CD are both parallel to EF if either of them is.

But if neither AB nor CD is parallel to EF,

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draw GM parallel to AB, and KN to CD, meeting BH and DL produced if necessary at M and N respectively,

let AB, CD produced if necessary meet EF at O and P respectively.

F

Then since AB is equal to CD

Нур. I. 29.

therefore GM is equal to KN.

Hence in the right-angled triangles MGH, NKL,

the side GM is equal to the side KN,

and the side GH is equal to the side KL,

Hyp.

therefore the angle MGH is equal to the angle NKL, I. 20, Cor. 1, therefore the angle AOG is equal to the angle CPK, I. 23, Cor. 2. that is, AB and CD make equal angles with EF.

Q.E.D.

THEOR. 34. If there are two pairs of straight lines all of which are parallel, and the intercepts made by each pair on a straight line that cuts them are equal, then the intercepts on any other straight line that cuts them are also equal.

Let AB, CD and EF, GH be two pairs of straight lines all of which are parallel, and let the intercepts AC, EG on the straight line AG be equal:

K

F

L

then shall the intercepts BD, FH on any other straight line BH

which cuts them be equal.

If BH is parallel to AG,

then BD is equal to AC,

I. 29.

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