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DEF. 24. A diagonal is the straight line joining the vertices of any
angles of a polygon which have not a common arm. DEF. 25. The perimeter of a rectilineal figure is the sum of its sides. DEF. 26. A quadrilateral is a polygon of four sides, a pentagon one
of five sides, a hexagon one of six sides, and so on. DEF. 27. A triangle is a figure contained by three straight lines. DEF. 28. Any side of a triangle may be called the base, and the
opposite angular point is then called the vertex. DEF. 29. An isosceles triangle is that which has two sides equal ;
the angle contained by those sides is called the vertical
angle, the third side the base. DEF. 30. A triangle which has one of its angles a right-angle is
called a right-angled triangle. A triangle which has one of its angles an obtuse angle is called an obtuse-angled triangle. A triangle which has all its angles acute is
called an acute-angled triangle. DEF. 31. The side of a right-angled triangle which is opposite to
the right-angle is called the hypotenuse. DEF. 32. The perpendicular to a given straight line from a given
point outside it is called the distance of the point from
the straight line. DEF. 33. Parallel straight lines are such as are in the same plane
and being produced to any length both ways do not meet. DEF. 34. When a straight line intersects two other straight lines
it makes with them eight angles, which have received special names in relation to the lines or to one another.
Thus in the figure 1, 2, 7, 8 are called exterior angles,
and 3, 4, 5, 6 interior angles ; again 4 and 6, 3 and 5, are called alternate angles ; lastly, I and 5, 2 and
6, 3 and 7, 4 and 8, are called corresponding ang'es. DEF. 35. A parallelogram is a quadrilateral whose opposite sides
are parallel. DEF. 36. A trapezium (or trapezoid) is a quadrilateral that has
only one pair of opposite sides parallel. DEF. 37. A parallelogram, one of whose angles is a right angle, is
called a rectangle. DEF. 38. A rhombus is a parallelogram that has all its sides
square is a rectangle that has all its sides equal. DEF. 40. The orthogonal projection of one straight line on another
straight line is the portion of the latter intercepted between perpendiculars let fall on it from the extremities of the
former. DEF. 41. A circle is a plane figure contained by one line, which is
called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is
called the centre of the circle. DEF. 42. A radius of a circle is a straight line drawn from the
centre to the circumference. DEF. 43. A diameter of a circle is a straight line drawn through
the centre and terminated both ways by the circumference. DEF. 44. If any and every point on a line, part of a line, or group
of lines (straight or curved), satisfies an assigned condition, and no other point does so, then that line, part of a line, or group of lines, is called the locus of the point satisfying that condition.
AXIOMS AND POSTULATES.
AXIOM 1. Magnitudes that can be made to coincide are equal.
AXIOM 2. Through two points there can be made to pass one, and only one, straight line ; and this may be indefinitely prolonged either way.
d. Any straight line may be made to fall on any other straight line with any given point on the one on any given point on the
B. Two straight lines which meet in one point cannot meet again, unless they coincide.
AXIOM 3. Through the same point there cannot be more than one straight line parallel to a given straight line.
POSTULATES OF CONSTRUCTION.
Let it be granted that
1. A straight line may be drawn from any one point to any other
point. 2. A terminated straight line may be produced to any length in a
straight line. 3. A circle may be drawn with any centre, with a radius equal to
any finite straight line.
EQUALITY OF AREAS.
DEF I. The altitude of a parallelogram with reference to a given
side as base is the perpendicular distance between the base
and the opposite side. DEF 2. The altitude of a triangle with reference to a given side as
base is the perpendicular distance between the base and the opposite vertex.
It follows from the General Axioms (d) and (e), as an extension of the Geometrical Axiom 1, that magnitudes which are either the sums or the differences of identically equal magnitudes are equal, although they may not be identically equal.
THEOR. I. Parallelograms on the same base and between the same parallels are equal.
Let ABCD, EBCF be two parallelograms on the same base BC, and between the same parallels AF, BC: