358. Corollary.-A diagonal divides a parallelogram into two equal triangles. But the diagonal does not divide the figure symmetrically, because the position of the sides of the triangles is reversed.

359. Theorem.-If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

Join AC. Then, the triangles ABC and CDA are equal. For the side AD is

A

B

mon. Therefore, the angles DAC and BCA are equal. But these angles are alternate with reference to the lines AD and BC, and the secant AC. Hence, AD and BC are parallel (130), and, for a similar reason, AB and DC are parallel. Therefore, the figure is a parallelogram.

360. Theorem.-If, in a quadrilateral, two opposite sides are equal and parallel, the figure is a parallelogram. If AD and BC are both equal and parallel, then AB is parallel to DC.

For, joining BD, the triangles thus formed are equal, since they have the side BD common, the side AD equal to

A

D

B

BC, and the angle ADB equal to its alternate DBC (284). Hence, the angle ABD is equal to BDC. But these are alternate with reference to the lines AB and DC, and the secant BD.

Therefore, AB and DC are parallel, and the figure is a parallelogram.

361. Theorem.-The diagonals of a parallelogram bisect each other.

The diagonals AC and BD are each divided into equa

the sides AB and CD equal (356), the angles ABH and CDH equal (125), and the angles BAH and DCH equal. Therefore, the triangles are equal (285), and AH is equal to CH, and BH to DH.

362. Theorem.-If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. To be demonstrated by the student.

RECTANGLE

363. If one angle of a parallelogram is right, the others must be right also (354).

A RECTANGLE is a right angled parallelogram. The rectangle has all the properties of other parallelo

grams, and the following peculiar to itself, which the student may demonstrate.

364. Theorem.-The diagonals of a rectangle are equal.

RHOMBUS.

365. When two adjacent sides of a parallelogram are equal, all its sides must be equal (356).

A RHOMBUS, or, as sometimes.

called, a LOZENGE, is a parallelogram having all its sides equal.

The rhombus has the follow

ing peculiarities, which may be demonstrated by the

student.

366. Theorem.-The diagonals of a rhombus are perpendicular to each other.

367. Theorem.-The diagonals of a rhombus bisect its angles.

SQUARE.

368. A SQUARE is a quadrilateral having its sides equal, and its angles right angles. The square may be shown to have all the properties of the parallelogram (359), of the rectangle, and of the rhombus.

369. Corollary. The rectangle and the square are the only parallelograms which can be inscribed in a circle (348).

EQUALITY.

370. Theorem.-Two parallelograms are equal when two adjacent sides and the included angle in the one, are respectively equal to those parts in the other.

For the remaining sides must be equal (356), and this becomes a case of Article 345.

371. Corollary.-Two rectangles are equal when two adjacent sides of the one, are respectively equal to those parts of the other.

372. Corollary.-Two squares are equal when a side of the one is equal to a side of the other.

APPLICATIONS.

373. The rectangle is the most frequently used of all quadrilaterals. The walls and floors of our apartments, doors and windows, books, paper, and many other articles, have this form.

Carpenters make an ingenious use of a geometrical principle in order to make their door and window-frames exactly rectangular. Having made the frame, with its sides equal and its ends equal,

they measure the two diagonals, and make the frame take such

a shape that these also will be equal.

In this operation, what principle is applied?

374. A rhombus inscribed in a rectangle is the basis of many ornaments used in architecture and other

work.

B

375. An instrument called parallel rulers, used in drawing parallel lines, consists of two rulers, connected by cross pieces with pins in their ends. The rulers may turn upon the pins, varying their distance. The distances between the pins along the rulers, that is, AB and CD, must be equal; also, along the cross pieces, that is, AC and BD. Then the rulers will always be parallel to each other. If one ruler be held fast while the other is moved, lines drawn along the edge of the other ruler, at different positions, will be parallel to each other.

C

What geometrical principles are involved in the use of this instrument?

EXERCISES.

376.-1. State the converse of each theorem that has been given in this chapter, and determine whether each of these converse propositions is true.

2. To construct a parallelogram when two adjacent sides and an angle are given.

3. What parts need be given for the construction of a rect angle?

4. What must be given for the construction of a square? 5. If the four middle points of the sides of any quadrilateral be joined by straight lines, those lines form a parallelogram.

6. If four points be taken, one in each side of a square, at equal distances from the four vertices, the figure formed by joining these successive points is a square.

7. Two parallelograms are similar when they have an angle in the one equal to an angle in the other, and these equal angles included between proportional sides.

MEASURE OF AREA.

377. The standard figure for the measure of surfaces is a square. That is, the unit of area is a square, the side of which is the unit of length, whatever be the extent of the latter.

Other figures might be, and sometimes are, used for this purpose; but the square has been almost universally adopted, because

1. Its form is regular and simple;

2. The two dimensions of the square, its length and breadth, are the same; and,

3. A plane surface can be entirely covered with equal squares.

The truth of the first two reasons is already known to the student that of the last will appear in the following theorem.

378. Any side of a polygon may be taken as the base.

The ALTITUDE of a parallelogram is the distance between the base and the opposite side. Hence, the altitude of a parallelogram may be taken in either of two

ways.

AREA OF RECTANGLES.

379. Theorem.. The area of a rectangle is measured by the product of its base by its altitude.

That is, if we multiply the number of units of length contained in the base, by the number of those units.