that is, AB is equal to DC, AD to BC, and the area ABC to the area ADC*.

Q. E. D.

COR. Hence if the adjacent sides of a parallelogram are equal, all its sides are equal.

Def. 41. A parallelogram all whose sides are equal is called a rhombus.

Def. 42. A square is a rectangle that has all its sides equal.

THEOREM 29.

If two parallelograms have two adjacent sides of the one respectively equal to two adjacent sides of the other, and likewise an angle of the one equal to an angle of the other; the parallelograms are identically equal.

Part. En. Let ABCD, EFGH be two parallelograms which have two adjoining sides AB, BC of the one equal respectively to two adjoining sides EF, FG of the other, and have likewise the included angles B and Fequal;

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it is required to prove that the parallelograms are identically equal.

Proof. For if the point B were placed on F, and the

line BC along the line FG;

then because BC = FG,

(Hyp.)

therefore the point C will fall on G;

and because the angle ABC= the angle EFG,

(Hyp.)

* Euclid, I. 34.

therefore BA will fall along FE;

and because BA == FE,

therefore the point A will fall on E.

(Hyp.)

(Hyp.)

(Ax. 5.)

And because AD is parallel to BC, therefore AD will fall along EH;

and similarly CD will fall along GH,

and therefore the point D will fall on the point H; that is, the parallelograms are identically equal.

COR.

Q. E. D.

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

THEOREM 30.

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

Part. En. Let ABCD be a quadrilateral in which the opposite sides AB, CD are equal and parallel;

B

it is required to prove that AC is equal and parallel to BD.

Proof. Join AD.

Then because AB is parallel to CD,

(Hyp.)

therefore the angle BAD is equal to the alternate angle

ADC;

(Th. 22.)

and therefore in the triangles BAD, CDA, we have

and the angle BDA = the angle CAD;

but these are alternate angles;

and therefore AC is parallel to BD*.

(Th. 21.) Q. E. D.

THEOREM 31.

Straight lines which are equal and parallel have equal projections on any other straight line; conversely, parallel straight lines which have equal projections on another straight line are equal; and equal straight lines, which have equal projections on another straight line, are equally inclined to that line.

B

E

a

Part. En. Let AB, CD be equal and parallel straight lines, and let ab, cd be their projections on any other straight line. Then shall ab be equal to cd.

Proof. Through A, C draw AE, CF parallel to abcd, meeting Bb, Dd in E, F.

* Euclid, I. 33.

Then because BA, AE, BE are respectively parallel to DC, CF, DF,

therefore the angle BAE = the angle DCF,

and the angle BEA = the angle DFC,

(Hyp.)

(Th. 27. Cor. 2.)

and the hypotenuse AB = the hypotenuse CD; (Hyp.)

If there are three parallel straight lines, and the intercepts made by them on any straight line that cuts them are equal, then the intercepts on any other straight line that cuts them are equal.

Let the three parallel straight lines AD, BE, CF make equal intercepts on the straight line AC, that is, let AB=BC.

Then shall the intercepts on any other line DEF be equal, that is, DE shall be equal to EF.