An Obtuse Angled Triangle has one of its angles obtuse.

An Acute Angled Triangle has all its angles acute.

The Three Angles of any Triangle taken together are equal to two right angles, or 180°.

The Difference of the Squares of two sides of a Triangle is equal to the product of their sum, and difference.

The Sides of a Triangle are proportional to the Sines of their opposite angles.

Quadrangles, or Quadrilaterals, are plane figures bounded by four right lines.

A Square is a quadrilateral having all its sides equal, and all its angles right angles. The Diagonal of a Square is equal to the square root of twice the square of its side: and the Side of the Square is equal to the square root of half the square of its diagonal.

The Diagonal is a right line drawn across a quadrilateral figure from one angle to another. The sum of the squares of the two diagonals of every parallelogram is equal to the sum of the squares of the four sides.

A Parallelogram is a quadrilateral whose opposite sides are parallel.

A Rectangle is a parallelogram having four right angles.
A Rhomboid is an oblique angled parallelogram.

A Rhombus, or Lozenge, is a quadrilateral, whose sides are all equal, but its angles oblique.

A Trapezium is a quadrilateral, which has none of its sides parallel to each other.

A Trapezoid is a quadrilateral, which has only two of its sides parallel.

sides.

Polygons are plane figures bounded by more than four

A Regular Polygon has all its sides, and angles equal.

The Perimeter of a figure is the sum of all its sides.

To Bisect is to divide into two equal parts.

To Trisect is to divide into three equal parts.

To Inscribe is to draw one figure within another, so that all the angles of the inner figure touch either the angles, sides, or planes, of the external figure.

To Circumscribe is to draw a figure round another, so that either the angles, sides, or planes, of the circumscribing figure touch all the angles of the figure within it.

LINES, ANGLES, AND FIGURES.

To divide a given right Line into Two equal Parts.

From the extremities of the line as centres, and with any opening in the compasses, greater than half the given line, as a radius, describe arcs intersecting each other above, and below the given line. A line being drawn through these intersections will divide the given line into two equal parts.

An Arc of a Circle is bisected in the same manner.

To bisect an Angle.

From the angular point measure equal distances on the two lines (forming the angle) and from these points with the same distance as radius describe arcs intersecting each other. A line drawn from their intersection to the angular point will bisect the angle.

To erect a Perpendicular.

From the point A set off any length 4 times to c, from A as a centre with 3 of those parts describe an arc at B, and from c with 5 of them cut the arc at B. Draw AB which will be the perpendicular required. Any equimultiples of these numbers, 3, 4, 5, may be used for erecting a perpendicular. Plate 2, HEIGHTS AND DISTANCES, and PRACTICAL GEOMETRY, Fig. 1.

To erect a Perpendicular.

Set off on each side of the point A any two equal distances AD, AE. From D and E as centres, and with any radius greater than half D E, describe two arcs intersecting each other in F. Through A and F draw the line AF, and it will be the Perpendicular required. Fig. 1, Plate PRACTICAL GEOMETRY.

To let fall a Perpendicular.

From D as a centre, and with any radius, describe an arc intersecting the given line. From the points of intersection C and E, with any radius, describe two arcs cutting each other at F. Through D and F draw a line, and DF will be the perpendicular required. Fig. 2.

To draw a Line parallel to a given Line.

From any point D in the given line with the radius D C, describe the arc CE, and from C with the same radius describe the arc D E.

D to F. Through c and F

required. Fig. 3.

Take E C, and set it off from draw C F for the Parallel

To divide an Angle into two equal Parts.

From B as a centre with any radius describe an arc a C. From A and C with any radius describe arcs intersecting each other in D. Then draw BD, and it will bisect the Angle. Fig. 4.

To divide a right Angle into three equal Parts.

From B as a centre with any radius describe the arc AC. From A with the radius A B cut the arc AC in D, and with the same radius from c cut it in E. Then through the

intersections D and E draw the lines BD, BE, and they will trisect, or divide the Angle into three equal parts. Fig. 5.

To find the Centre of a Circle.

Draw any chord AB, and bisect it by the perpendicular Divide CD into two equal parts, and the point of bisection o will be the Centre required. Fig. 6.

CD.