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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 solu
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.
is called plus: it indicates addition.
The sign 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 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 A× 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 Ax (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 AB2; 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 means the the product of A and B.
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 the 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 A 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 an-, gles, 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 ana the same straight line.
pose, however, that on being produced, these lines begin to separate at C, the one becoming CD, the other CE. From the point C draw the line CF, making with AC the right angle ACF. Now, since ACD is a straight line, the angle FCD will be a right angle (Prop. I. Cor. 1.); and since ACE is a straight line, the angle FCE will likewise be a right angle. Hence, the angle FCD is equal to the angel FCE (Ax. 10.); which can only be the case when the lines CD and CE coincide: therefore, the straight lines which have two points A and B common, cannot separate at any point, when produced; hence they form one and the same straight line.
PROPOSITION III. THEOREM.
If a straight line meet two other straight lines at a common point, making the sum of the two adjacent angles equal to two right angles, the two straight lines which are met, will form one and the same straight line.
Let the straight line CD meet the two lines AC, CB, at their common point C, making the sum of the two adjacent angles DCA, DCB, equal to A two right angles; then will CB be the prolongation of AC, or AC and CB will form one and the same straight line.
For, if CB is not the prolongation of AC, let CE be that prolongation: then the line ACE being straight, the sum of the angles ACD, DCE, will be equal to two right angles (Prop. I.). But by hypothesis, the sum of the angles ACD, DCB, is also equal to two right angles: therefore, ACD+DCE must be equal to ACD+DCB; and taking away the angle ACD from each, there remains the angle DCE equal to the angle DCB, which can only be the case when the lines CE and CB coincide; hence, AC, CB, form one and the same straight line.
PROPOSITION IV. THEOREM.
When two straight lines intersect each other, the opposite or vertical angles, which they form, are equal.