35. Theorem. lying in the same

they are parallel.

Equal Oblique Lines.

Reciprocally, if two straight lines, plane, are perpendicular to a third,

Proof. For the line, drawn through the point M parallel to CD, must be perpendicular to EF, and must therefore coincide with AB.

36. Theorem. If two straight lines, as AB, CD (fig. 15), are parallel to a third, EF, they are parallel to each other.

Proof. For, by the definition of parallel lines, they have the same direction with this third, and are therefore parallel.

CHAPTER VI.

PERPENDICULAR AND OBLIQUE LINES.

37. Theorem. Only one perpendicular can be drawn from a point to a straight line.

Proof. For, if two perpendiculars are erected in the same plane, at two different points, M and P (fig. 16) of the line AB, they are parallel, by § 35, and cannot meet at any point, as C.

38. Theorem. Two oblique lines, as CE and CF (fig. 17), drawn from the point C to the line AB, at equal distances DE and DF from the perpendicular CD, are equal.

Shortest Distance from a Line.

Proof. For, if CDB be folded over upon CDA, DB will fall upon DA, because the right angles CDB and CDA are equal; the point F will fall upon E, because DF and DE are equal; and the straight lines CF and CE will coincide.

39. Theorem.

A perpendicular measures the short

est distance of a point from a straight line.

Proof. Let the perpendicular CD (fig. 18) and the oblique line CF be drawn from the point C to the line AB. Produce CD to DE, making DE equal to DC, and join FE, we shall, by § 18, have

But

and

CE<FC+ FE.

CE=2 CD,

FC+FE-2 FC,

for FC and FE are equal, because they are oblique lines drawn from the point F to the line CE at equal distances DC and DE from the perpendicular.

40. Lemma. The sum of two lines, as CA and CB (fig. 19), drawn to the extremities of the line AB, is greater than that of two other lines DA and DB, similarly drawn, but included by them.

Proof. Produce DA to E.

We have, by § 18,

and

AC+ CE> AD + DE,

DEBE > DB.

Oblique Lines unequally Distant from the Perpendicular.

The sum of these inequalities is

AC + CE+DE+ BE > AD + DE+ DB, or, striking out the common term DE, and substituting for CEBE, its equal BC,

AC+BC > AD + DB.

41. Theorem. Of two oblique lines, CF and CG (fig. 18), drawn unequally distant from the perpendicular, the more remote is the greater.

Proof. For, the figure being constructed as in § 39, and GE being joined, we have, by the preceding proposition,

or, as in § 39,

and

GC GE> FC + FE;

2 GC 2 FC,

GC FC.

42. Theorem. If from the point C the middle of the straight line AB (fig. 20), a perpendicular EC be drawn:

1. Any point in the perpendicular EC is equally distant from the two extremities of the line AB.

2. Any point without the perpendicular, as F, is at unequal distances from the same extremities A and B.

Proof. 1. The distances EA and EB are equal, since they are oblique lines drawn at equal distances CA and CB from the perpendicular AB.

2. The distance FA is greater than FB; for

Polygon, Triangle, Square.

CHAPTER VII.

SIDES AND ANGLES OF POLYGONS.

43. Definitions. A plane figure is a plane terminated on all sides by lines.

If the lines are straight, the space which they contain is called a rectilineal figure, or polygon (fig. 21), and the sum of the bounding lines is the perimeter of the polygon.

44. Definitions. The polygon of three sides is the most simple of these figures, and is called a triangle ; that of four sides is called a quadrilateral; that of five sides, a pentagon; that of six, a hexagon, &c.

45. Definitions. A triangle is denominated equilateral (fig. 22), when the three sides are equal, isosceles (fig. 23), when two only of its sides are equal, and scalene (fig. 24), when no two of its sides are equal.

46. Definitions. A right-triangle is that which has a right angle. The side opposite to the right angle is called the hypothenuse. Thus ABC (fig. 25) is a triangle right-angled at A, and the side BC is the hypothenuse.

47. Definitions. Among quadrilateral figures, we distinguish

The square (fig. 26), which has its sides equal, and its angles right angles. (See § 73).

Rectangle, Parallelogram, Rhombus, Trapezoid, Diagonal.

The rectangle (fig. 27), which has its angles right angles, without having its sides equal.

The parallelogram (fig. 28), which has its opposite sides parallel.

The rhombus or lozenge (fig. 29), which has its sides equal without having its angles right angles.

The trapezoid (fig. 30), which has two only of its sides parallel.

48. Definition. A diagonal is a line which joins the vertices of two angles not adjacent, as AC (fig. 30.)

49. Definitions. An equilateral polygon is one which has all its sides equal; an equiangular polygon is one which has all its angles equal.

50. Definition. Two polygons are equilateral with respect to each other, when they have their sides equal, each to each, and placed in the same order; that is, when, by proceeding round in the same direction, the first in the one is equal to the first in the other, the second in the one to the second in the other, and so on.

In a similar sense are to be understood two polygons equiangular with respect to each other.

The equal sides in the first case, and the equal angles in the second, are called homologous.

51. Theorem. Two triangles are equal, when two sides and the included angle of the one are respectively equal to two sides and the included angle of the other.

Proof. In the two triangles ABC, DEF, (fig. 31), let the angle A be equal to the angle D, and the sides AB, AC, respectively equal to DE, DF.