7*12
THEORY AND PRACTICE OF ENGINEERING.
Book II.
The three internal angles of every triangle are equal totwo right angles, or to 180 degrees. From the summit ofthe angles NOP describe circles, each divided into 360 de-grees, and if we add the degrees contained between thelines of the three angles, as P N O 68, NOP 60, O P N 52,we shall find the contents of the three angles together 180degrees, or the double of two right angles.
A Common Triangle is that which is comprised betweentwo triangles, of which it contains an equal portion, andwhich has for its base the same as that of the two tri-angles comprised between the same parallels. The triangleGHI is common with respect to YIII and ZI H, becauseit is comprised between two triangles
Figures of Four Sides, or Quadrilaterals : —
A Square has four equal sides, and four right angles, asA BCD.
I-ig. C62.
N G
Fig. (363.
In the rectangle A B C D, the side B C is parallel to theside A D, and the side A B parallel to the side D C. Theline A B is perpendicular to the two lines B C, A D, thetwo other lines are therefore parallel: in like manner theline A D is perpendicular to the two lines A B, DC; thetwo lines A B, DC, are therefore parallel.
A Parallelogram has four right angles, but its sides areunequal, two being shorter than the others, as EFGII.
The opposite sides of rectangles are equal; and a linefalling upon parallel lines, as we have seen, make the alter-nate angles equal.
By superposition the relative proportions of the squareand parallelogram can be ascertained ; this method wasvery much used by the ancient geometricians : when twofigures so applied are found to coincide and to fill up thesame space, we infer that they are equal each to each.
When Euclid endeavoured to prove that two triangleswhich have two sides of the one equal to two sides of theother, and also the angles contained by those sides, equal, hesupposes one triangle to be placed over the other : on suclia principle we compare rectilineal figures, for if it be shownthat the square when placed over the parallelogram occupiesonly two-thirds of that figure, we infer that it requires halfentirely its area in addition to cover it. It is easily de-monstrated also that any two equal rectilineal figures may,by resolving them into parts, be applied by superpositionone above the other, so as entirely to agree in quantity.
A parallelogram is bisected by each of its diagonals, forthe triangles into which it is divided are equal to one another:and, consequently, if one angle of a parallelogram be a rightangle, all its angles will be right angles. Hence we learn thata rhombus has all its sides equal to one another; that a rect-angle has all its angles right angles; and that a square hasall its sides equal, and all its angles right angles.
Euclid has clearly shown that the opposite sides andangles of parallelograms are equal, and that their diagonalsbisect one another ; and, conversely, if in any quadrilateralfigure, the opposite sides be equal, or if the opposite anglesbe equal, or if the diagonals bisect one another, that quad-rilateral shall be a parallelogram. The same writer has alsoproved that the complements of the parallelograms whichare about the diagonals of any parallelogram are equal toone another.
A Rhombus has its four sides equal, but not at rightangles, as KLM N.
The rhombus has the peculiar property of its diagonalsangles; and therefore whenever a quadrilateral has all itsbisect one another at right angles, it is a rhombus.
A Rhomboid has its opposite sides and angles equal, as O P Q R, without being equi-lateral or rectangular. The diagonals of all quadrilaterals are straight lines, which jointhe opposite angles, and consequently would divide the figure into four triangles.
Fig. 606.