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3. Mark off LN equal to GK. Join ON, and from A and

B, draw AP, BQ parallel to ON (Pr. 9). The divisions L, 2, N, P, are proportional to the divisions L, B, 0, A (Pr. 16).

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4. Make GR equal to LQ, and KS equal to NP. Then the

divisions from G to S are in the same proportion as the divisions from L to A. Hence, if we draw the lines AS, BR, they will converge to the same point as the given lines CD, EF.

Section 11.-TRIANGLES.

DEFINITIONS.

(Hitherto we have treated only of LINES and ANGLES.)

1. “A figure is that which is inclosed by one or more boundaries”

(Euc. I. Def. 14).

2. “Rectilineal figures are those which are contained by straight lines” (Euc. I. Def. 20).

NOTE.—Two straight lines cannot inclose a space ” (Euc. I. Ax. 10). Hence the triangle is the most simple of all rectilineal figures.

3. A triangle is a figure which is bounded by three straight lines.

Ex. ABC

B

Note 1.-It is therefore called a trilateral, or three-sided figure.

NOTE 2. As every rectilineal figure contains as many angles as sides, it is often named from its anglés or corners. Hence the term triangle.

NOTE 3.-Triangles are of six kinds. Three of these are named from the comparative lengths of their sides, and three from the sizes of their angles.

4. An equilateral triangle is that which has three equal sides.

Ex, ABC

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5. An isosceles triangle is that which has only two sides equal.

Ex. ABC

A

B

6. A scalene triangle is that which has three unequal sides. E.c.

ABC

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B

С

NOTE.— The above terms have reference to the sides of the triangle.

7. A right-angled triangle is that which has a right angle. Ex.

ABC

с

B

NOTE.—The side which is opposite the right angle (AC) is called the hypotenuse. The other sides are called the base, and perpendicular, irrespective of the position of the figure.

8. An obtuse-angled triangle is that which has one of its angles an

obtuse angle. Ex. ABC

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9. An acute-angled triangle is that which has three acute angles.

Ex. ABC

A

B

NOTE.— The above terms have reference to the angles of the triangle.

10. The vertex (plural vertices) of a triangle is its highest angle.

Ex. A

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NOTE.—It is also called the apex, or the vertical angle.

11. The base of a triangle is generally its lowest side. Ex. AB

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NOTE.—Both in an isosceles triangle and in a right-angled triangle, the position of the base is changed.

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12. The altitude of a triangle is its perpendicular height, i.e., the

length of a perpendicular drawn from the vertex to the base, or to the base produced. Ex. AB

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13. The perimeter of a figure is its whole boundary. Thus, if one

side of an equilateral triangle be 5, its perimeter is 15.

14. A chord is any straight line drawn across a circle, provided that

it does not pass through the centre. Ex. AB

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To construct an equilateral triangle on a given base AB,

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1. From centre A, with radius AB, describe an arc BD.

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