D

A

An angle is usually denoted by three letters, the middle letter being that at the angular point. Thus, the angle formed by the lines AC, BC is the angle ACB. And the angle formed by the lines DC, BC, is the angle DC. Therefore the magnitude or opening of an angle is not dependant on the lengths of the lines which include or make the angle: thus, DC is less than AC, but the angle DCB is greater than the angle ACB; or the inclination of the line DC to BC is greater than the inclination of AC to BC.

B

Scholium. From the foregoing definition of an angle, it fullows, that if two straight lines in the same plane are not inclined to each other, they cannot form an angle, and consequently cau never be produced so as to meet, in which case the lines are said to be parallel: Therefore,

9. Parallel straight lines are such as are in the same plane but not inclined to each other, A or when indefinitely produced both ways do never meet, as AB, CD.

-B

C

-D

E

A

B

IG

H

C

D

F

So if two straight lines AB, CD intersect a third straight line EF (all in the same plane) and are equally inclined to that line, or make the angles AGE, CHE equal, the two lines have no inclination to one another, but are parallel or equidistant; and when all the angles at G and H are equal to each other, the line GH is the distance of those parallels.

10. A right angle is formed by two lines which are perpendicular to each other. Thus if

P

PQ is perpendicular to RS, each of the angles Q
PQR, PQS is a right angle.

G

S

11. An acute angle is less than a right angle; as the angle GQS.

12.

An obtuse angle is greater than a right angle; as the angle GQR. Those are called oblique angles.

13. The sides of a right lined plane figure are straight lines.

14

When the number of sides are three, the figure is a triangle.

15. An equilateral triangle is that whose sides are all equal, as A.

16. An isosceles triangle is that which has only two sides equal, as B.

17. A scalene triangle is when all the three sides are unequal, as C.

18. A right angled triangle is that which has one right angle, as D.

19. An acute angled triangle has all its angles acute, as E.

20. An obtuse angled triangle has one obtuse angle, as F.

21. Every plane figure bounded by four right lines is called a quadrangle or quadrilateral. And when the opposite sides are respectively parallel, the quadrilateral is called a parallelogram.

22. A rectangle is a parallelogram having all its angles right ones, as G.

23. A square is a parallelogram having all its sides equal, and all its angles right ones, as H.

24. A rhomboid is an oblique angled parallelogram, as I.

25. A rhombus is an equilateral rhomboid

I

G

H

as K.

K

26.

A trapezoid is a quadrilateral with only two parallel sides, as L.

27. A trapezium is a quadrilateral in which none of the sides are parallel, as M.

28.

A right line joining any two opposite N angles of a quadrilateral is called a diagonal, as NO.

29. The side PQ upon which any parallelogram PQRS, or triangle PSQ, is supposed to stand, is called the base; and the perpendicular ST, falling thereon from the opposite angle at S, is called the height or altitude of the parallelogram or triangle.

S

P T

L

M

Ο

R

The perpendicular ST is also called the distance of the point S from the line PQ, or the distance of the parallels SR, PQ.

30. All right lined plane figures having more than four sides are generally called polygons. And a regular polygon is one whose angles as well as sides are all equal.

AXIOMS.

31. THINGS which are equal to the same thing, or to equal things, are equal to each other.

32.

If equals are added to equals, the wholes are equal.

33. If equals are subtracted from equals, the remainders are equal.

34. Every whole is equal to all its parts taken together.

35. Things which are the like parts of the same thing, are equal.

36. Magnitudes which coincide with one another, that is which exactly fill the same space, are identical, or mutually equal in all their parts.

37. All right angles are equal to one another.

N. B. A Proposition is something either proposed to be done, or to be demonstrated, and is either a problem or a theorem.

A Problem is something proposed to be done.

A Theorem is something proposed to be demonstrated.

A Corollary is a consequent truth gained from some preceding truth or demonstration.

A Scholium is a remark or observation made upon something going before it.

A Lemma is something premised or demonstrated, in order ta render what follows more easy.

OF THE ANGLES OF RIGHT-LINED PLANE FIGURES.

THEOREMS.

38. If there be two triangles ABD, abd, having two sides BA, BD of one triangle, respectively equal to two sides ba, bd of the other, and the included angles B and b also equal; the triangles are identical, or equal in all respects.

If we conceive the triangle ABD to be so applied to the

B

triangle ald that the angle B

a

may coincide with the angle b,

and the side BA fall upon ba: Then the angles at B and b be

ing supposed equal, the side BD will fall upon bd, and the point D on d; consequently AD will coincide with ad: hence it is manifest that the triangles are identical or equal in all respects; and therefore AD will be equal to ad, and the adjacent angles A, and D, equal to the angles a, and d, respectively.

And in a similar manner it is proved that triangles are identical when the bases (AD, ad) and the adjacent angles (A, D; a, d) are equal.-For if one triangle is supposed to be placed upon the other so that the bases, and adjacent angles coincide, the other sides, and also the two vertical angles, must coincide, and will therefore be respectively equal.

38. The angles which one right line make with another on the same side, ae together equal to two right angles.

Let the line DP meet the line AB in the point P, then the two angles DPB, DPA are together equal to two right angles.

A P B

If the angles are equal, each will be equal to a right angle (10).

But when they are unequal, let PC be perpendicular to AB.

Then the three angles BPD, DPC, CPA, together are equal to two right angles (34).

But the two angles DPC, CPA are together equal to the angle DPA.

Therefore the two angles DPB, DPA together make two right angles.

Corol. 1. Hence it appears that all the angles at the same point (P) on the same side of a right line (AB) are together equal to two right angles. And consequently all the angles that can be made round a given point (P) are equal to four right ones.