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9. When two straight lines, AB, AC, meet each other, their inclination or opening is called an angle, which is greater or less as the lines are more or less inclined or opened. The point of intersection A is the vertex of the angle, and the lines AB, AC, are its sides.
The angle is sometimes designated simply by the letter at the vertex A; sometimes by the three letters BAC, or CAB, the letter at the vertex being always placed in the middle.
Angles, like all other quantities, are susceptible of addition, subtraction, multiplication, and division.
Thus the angle DCE is the sum of the two angles DCB, BCE; and the angle DCB is the difference of the two A angles DCE, BCE.
10. When a straight line AB meets another straight line CD, so as to make the adjacent angles BAC, BAD, equal to each other, each of these angles is called a right angle; and the line AB is said to be perpendicular to CD.
11. Every angle BAC, less than a right angle, is an acute angle; and every angle DEF, greater than a right angle, is an obtuse angle.
12. Two lines are said to be parallel, when being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced.
3. A plane figure is a plane terminated on all sides by lines, either straight or curved.
If the lines are straight, the space they enclose is called a rectilineal figure, or polygon, and the lines themselves, taken together, form the contour, or perimeter of the polygon.
14. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadrilateral; that of five, a pentagon; that of six, a hexagon; that of seven, a heptagon: that of eight, an octagon; that of nine, a nonagon; that of ten, a decagon; and that of twelve, a dodecagon.
15. An equilateral triangle is one which has its three sides equal; an isosceles triangle, one which has two of its sides equal; a scalene triangle, one which has its three sides unequal. 16. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse. Thus, in the triangle ABC, right-angled at A, the side BC is the hypothenuse.
17. Among the quadrilaterals, we distinguish :
The square, which has its sides equal, and its angles right-angles.
The rectangle, which has its angles right angles, without having its sides equal.
The parallelogram, or rhomboid, which has its opposite sides parallel.
'The rhombus, or lozenge, which has its sides equal, without having its angles right angles.
And lastly, the trapezoid, only two of whose sides are parallel.
18. A diagonal is a line which joins the vertices of two angles not adjacent to each other. Thus, AF, AE. AD, AC, are diagonals.
19. An equilateral polygon is one which has all its sides equal; an equiangular polygon, one which has all its angles equal.
20. Two polygons are mutually equilateral, when they have their sides equal each to each, and placed in the same order
that is to say, when following their perimeters in the same direction, the first side of the one is equal to the first side of the other, the second of the one to the second of the other, the third to the third, and so on. The phrase, mutually equiangular, has a corresponding signification, with respect to the angles.
In both cases, the equal sides, or the equal angles, are named homologous sides or angles.
Definitions of terms employed in Geometry.
An axiom is a self-evident proposition.
A theorem is a truth, which becomes evident by means of a train of reasoning called a demonstration.
A problem is a question proposed, which requires a solution.
A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem.
The common name, proposition, is applied indifferently, to theorems, problems, and lemmas.
A corollary is an obvious consequence, deduced from one or several propositions.
A scholium is a remark on one or several preceding propositions, which tends to point out their connexion, their use, their restriction, or their extension.
A hypothesis is a supposition, made either in the enunciation of a proposition, or in the course of a demonstration.
Explanation of the symbols to be employed.
The sign is the sign of equality; thus, the expression A=B, signifies that A is equal to B.
To signify that A is smaller than B, the expression A<B is used.
To signify that A is greater than B, the expression A>B is used; the smaller quantity being always at the vertex of the angle.
The sign is called plus: it indicates addition. The sign is called minus: it indicates subtraction. Thus, A+B, represents the sum of the quantities A and B; A-B represents their difference, or what remains after B is taken from A; and A-B+C, or A+C-B, signifies that A and C are to be added together, and that B is to be subtracted from their sum.
The sign x indicates multiplication: thus, Ax B represents the product of A and B. Instead of the sign x, a point is sometimes employed; thus, A.B is the same thing as Ax B. The same product is also designated without any intermediate sign, by AB; but this expression should not be employed, when there is any danger of confounding it with that of the line AB, which expresses the distance between the points A and B.
The expression A× (B+C-D) represents the product of A by the quantity B+C-D. If A+B were to be multiplied by A-B+C, the product would be indicated thus, (A+B) × (A-B+C), whatever is enclosed within the curved lines, being considered as a single quantity.
A number placed before a line, or a quantity, serves as a multiplier to that line or quantity; thus, 3AB signifies that the line AB is taken three times; A signifies the half of the angle A.
The square of the line AB is designated by AB'; its cube by AB3. What is meant by the square and cube of a line, will be explained in its proper place.
The sign indicates a root to be extracted; thus √2 means the square-root of 2; √A×B the product of A and B.
means the square-root of
1. Things which are equal to the same thing, are equal to each other.
2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal.
4. If equals be added to unequals, the wholes will be unequal.
5. If equals be taken from unequals, the remainders will be unequal.
6. Things which are double of the same thing, are equal to each other.
7. Things which are halves of the same thing, are equal to each other.
8. The whole is greater than any of its parts.
9. The whole is equal to the sum of all its parts.
10. All right angles are equal to each other.
11 From one point to another only one straight line can be drawn.
12. Through the same point, only one straight line can be drawn which shall be parallel to a given line.
13. Magnitudes, which being applied to each other, coincide throughout their whole extent, are equal.
PROPOSITION I. THEOREM.
If one straight line meet another straight line, the sum of the two adjacent angles will be equal to two right angles.
Let the straight line DC meet the straight line AB at C, then will the angle ACD + the angle DCB, be equal to two right angles.
At the point C, erect CE perpendicular to AB. The angle ACD is the sum of the angles ACE, ECD: therefore ACD+DCB is the sum of the three angles ACE, ECD, DCB: but he first of these three angles is a right angle, and the other two make up the right angle ECB; hence, the sum of the two angles ACD and DCB, is equal to two right angles.
Cor. 1. If one of the angles ACD, DCB, is a right angle. the other must be a right angle also.
Cor. 2. If the line DE is perpendicular to AB, reciprocally, AB will be perpendicular to DE.
For, since DE is perpendicular to AB, the ▲ angle ACD must be equal to its adjacent angle DCB, and both of them must be right angles (Def. 10.). But since ACD is a right angle, its adjacent angle ACE must also be a right angle (Cor. 1.). Hence the angle ACD is equal to the angle ACE, (Ax. 10.) therefore AB is perpendicular to DE.
Cor. 3. The sum of all the successive angles, BAC, CAD, DAE, EAF, formed on the same side of the straight line BF, is equal to two right angles; for their sum is equal to that of the two adjacent angles, BAC, CAF.
Let A and B be the two common points. In the first place it is evident. that the two lines must coincide entirely between A and B, for otherwise there would be two straight lines between A and B, which is impossible (Ax. 11). Sup
PROPOSITION II. THEOREM.
Two straight lines, which have two points common, coincide with each other throughout their whole extent, and form one and the same straight line