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the other two sides its legs, and the line joining the middle points of the legs is called the median.

175. A trapezoid is called an isosceles trapezoid when its legs are equal.

176. The altitude of a parallelogram or trapezoid is the perpendicular distance between its bases.

177. The diagonal of a quadrilateral is a straight line joining two opposite vertices.

PROPOSITION XXXVII. THEOREM.

178. The diagonal of a parallelogram divides the figure into two equal triangles.

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Let ABCE be a parallelogram and AC its diagonal.

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(having a side and two adj. 4 of the one equal respectively to a side and

two adj. of the other).

Q. E. D.

179. In a parallelogram the opposite sides ar and the opposite angles are equal.

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Also,

BC AE, and AB= EC,

LB LE, and Z BAE= LBCE.

Draw AC

▲ ABC=▲ AEC,

(the diagonal of a □ divides the figure into two equal ▲

.. BC= AE, and AB = CE,

(being homologous sides of equal §).

LB LE, and Z BAE=

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BCE,

(having their sides || and extending in opposite directions fr their vertices).

180. COR. Parallel lines comprehended between para are equal.

181. COR. 2. Two parallel lines are everywhere equally distant. For if AB and DC are parallel,

A

D

Is dropped from any points in AB to DC, measure the d of these points from DC. But these Is are equal, by hence, all points in AB are equidistant from DC.

182. If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram.

B

Let the figure ABCE be a quadrilateral, having the side AE equal and parallel to BC.

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(having two sides and the included 4 of the one equal respectively to two sides and the included ▲ of the other).

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(when two straight lines are cut by a third straight line, if the alt.-int.

are equal, the lines are parallel).

.. the figure ABCE is a □,

§ 168

183. If the opposite sides of a quadrilat equal, the figure is a parallelogram.

B

E

Let the figure ABCE be a quadrilateral hav. AE and AB = EC.

To prove figure ABCE a □.

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(having three sides of the one equal respectively to three sides of

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(when two straight lines lying in the same plane are cut by a thi line, if the alt.-int. & are equal, the lines are parallel)

.. the figure ABCE is a □,
(having its opposite sides parallel).

184. The diagonals of a parallelogram bisect each other.

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Let the figure ABCE be a parallelogram, and let the diagonals AC and BE cut each other at 0.

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(having a side and two adj. 4 of the one equal respectively to a side and

two adj. of the other).

.. AO OC, and BO:

==

OE,

(being homologous sides of equal §).

Q. E. D.

Ex. 25. If the diagonals of a quadrilateral bisect each other, the figure

is a parallelogram.

Ex. 26. The diagonals of a rectangle are equal.

Ex. 27. If the diagonals of a parallelogram are

equal, the figure is a rectangle.

B

Ex. 28. The diagonals of a rhombus are perpendicular to each other, and bisect the angles of the rhombus.

Ex. 29. The diagonals of a square are perpendicular to each other,

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