If

edge of the board, with the guide pressed against one face and the point against the other. In the same manner any number of parallels may be drawn. the key be not well fastened, the instrument soon becomes disarranged. It may, however, be easily tested by reference to the point of departure of the line.

It will readily be seen how invaluable the gauge is for drawing long parallels, and particularly such as are required to be at small distances apart.

The same instrument, constructed of metal, with a few modifications tending to increase its accuracy, is used by locksmiths and other workers in metal.

110. The reasons for the frequent use of parallels in building constructions and the arts, are the same as those noticed in connection with the right-angle. Lines parallel to one another abound in almost every construction ; for example, in the sides of houses, the panels of doors, and in the sashes of windows, and in the great variety of objects of uniform breadth, such as beams, bricks, metal bars, rails, etc., etc.

Quadrilaterals.

III. Any plane figure bounded by four straight lines is termed a quadrilateral (fig. 92). A line joining two opposite angles of a quadrilateral is termed a diagonal.

112. A parallelogram is a quadrilateral having its opposite sides parallel (fig. 93).

113. A rhombus is a parallelogram which has all its sides equal, but not all its angles equal (fig. 94).

Fig. 97.

114. A rectangle has all its angles right (fig. 95). 115. A square is an equilateral rectangle (fig. 96).

116. A trapezoid is a quadrilateral having only two sides parallel (fig. 97).

117. A right-angled trapezoid has two angles right-angles. 118. Figure 98 represents a tessellated pavement made up of squares, parallelograms, and rectangular trapezoids.

Fig. 98.

In every parallelogram the opposite sides and angles are equal; and, conversely, if the opposite sides or the opposite angles of a quadrilateral are equal it is a parallelogram.

119. The triangles C D B, A B D, have the alternate angles CD B, ABD equal, A D B, DBC equal, and the side B D common to both so

that if the triangle C D B were turned round without being turned over, and C placed on A, D on B, and B on D, the triangles would coincide. Therefore B C

Fig. 99.

B

equals A D, A B equals C D, the angle C equals the angle A, and the sum of the angles at B equals the sum of the angles at A.

I 20.

Conversely, If the opposite sides are equal the figure is a parallelogram. For the three sides of CD B are respectively equal to the three sides of A B D, and therefore these triangles are equal, and the alternate angles C D B, A B D, are equal, and so also are the alternate angles A D B, CBD. Therefore CD is parallel to A B, and C B to A D.

DIAGONALS.

The diagonals of a parallelogram bisect each other.

121. For the vertically opposite triangles DOC, BOA are equal, having a side CD equal to a side A B, the alternate angles CDO, A BO, equal, and also the alternate angles D CO, BA O, equal.

122. If in a quadrilateral the two diagonals bisect each other, the figure is a parallelogram.

For the vertically opposite triangles, CO D, A O B, are then equal, since they have DO, OC, equal respectively to BO, OA, and the included angles at O equal. Thus D C is equal to B A.

R

Fig. 100.

Si

milarly, A D is equal to C B, and therefore the figure is a parallelogram (fig. 100).

Any line drawn through the centre 0 to the opposite side is bisected in 0.

Hence the triangles are equal, and ON is equal to OM (fig. 101).

If through the centre of a parallelogram two straight lines are drawn to opposite sides, the figure formed by joining their extremities will be a parallelogram. For these lines will bisect each other.

To construct a parallelogram, having given two sides and the included angle.

124. Make an angle equal to the given angle, and cut from its sides lengths equal to the two given sides. Through the extremities draw lines parallel to the sides of the angle.

The sum of the angles of a parallelogram are together equal to four right-angles.

F

D

E

125. Let the side D C be produced in one direction to E, and in the other to F. The sum of the adjacent angles at D is equal to two right angles, and so is the sum of the angles at C. But the angle FDA is equal to the alternate angle

B

Fig. 102.

BAD, and the angle ECB to the alternate angle A B C, and therefore the four angles of the parallelogram are equal to the sum of the angles at D and C, and are therefore equal to four right angles.

From this construction it follows that

The angles of a triangle are together equal to two right-angles.

126. Let A B D be the triangle, and through D draw E F parallel to A B (fig. 102). The three angles at D which are equal to two right-angles, are respectively equal to the three angles of the triangle.

It follows that in a right-angled triangle two angles

Fig. 103.

are acute, and their sum is a rightangle.

127. The exterior angle of a triangle is equal to the sum of the interior and opposite angles.

When the side of a triangle is

produced, the new angle formed is termed the ex

B

C

Fig. 104.

terior angle. It is the supplement* of the angle of the triangle which is adjacent to it, and is thereDfore equal to the other two angles, which are termed

interior and opposite (fig. 104).

To trisect a given angle.

128. Let A B C be the given angle. With centre B, and any radius, describe a semicircle, A CE. On a rule

E

A

or the straight edge of a sheet of paper mark off this radius. Place the edge of the paper so that it passes through C and has one extremity of the radius marked on the line A B produced as at D, while the other is on the circumference at E for example. Draw the line CED and the angle CD B, will be the third of CBA (fig. 105).

B

Fig. 105.

For the exterior angle A B C is equal to C + D. But since BC is equal to B E and to E D, the angle BCE is equal to BE C, and this to E B D plus BDE, that is, to twice the angle D. Therefore the angle

A B C is equal to three times the angle D.

By means of this proposition we may construct a trisector, as represented in the figure (fig. 106). The parts indicated move on rivets at D, B, E, and C;

* The supplement of an angle is that which must be added to it to make the sum equal to two right-angles.