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are all equal; of the angles D, E, F, that with the shortest arms is the largest, and that with the longest arms is the smallest.

8. If three straight lines are drawn from the same point, three different angles are formed.


Thus AB, AC, AD, drawn from A,

form the three angles BAC, CAD, A


The angles BAC, CAD, which

have a common arm AC, and lie on


opposite sides of it, are called adjacent angles; and the angle BAD, which is equal to angle BAC and angle CAD added together, is called the sum of the angles BAC and CAD. Since the angle BAD is obtained by adding together the two angles BAC and CAD, the angle CAD will be obtained by subtracting the angle BAC from the angle BAD; and similarly the angle BAC will be obtained by subtracting the angle CAD from the angle BAD. Hence the angle CAD is called the difference of the angles BAD and BAC; and the angle BAC is called the difference of the angles BAD and CAD.

9. The bisector of an angle is the straight line that divides it into two equal angles.

Thus (see preceding fig.), if angle BAC is equal to angle CAD, AC is called the bisector of angle BAD.

The word bisect, in Mathematics, means always, to cut into two equal parts.

10. When a straight line stands on another straight line, and makes the adjacent angles equal to each other, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.


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Thus, if AB stands on CD in such a manner that the adjacent angles ABC, ABD are equal to one another, then

these angles are called right angles, and AB is said to be perpendicular to CD.

11. An obtuse angle is one which is greater than a right angle.

Thus A is an obtuse angle.


12. An acute angle is one which is less than a right angle.

Thus B is an acute angle.


13. When two straight lines intersect each other, the opposite angles are called vertically opposite angles.

Thus AEC and BED are vertically opposite angles; and so are AED and BEC.





14. Parallel straight lines are such as

same plane, and being produced

ever so far both ways do not meet.


are in the

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Thus AB and CD are parallel straight lines.


If a straight line EF intersect two parallel straight lines AB, CD, the angles AGH, GHD are called alternate angles, and so are angles BGH, GHC; angles AGE, BGE, CHF, DHF are called exterior angles, and the interior opposite angles corresponding to these are CHG, DHG, AGH, BGH.

15. A figure is that which is inclosed by one or more boundaries; and a plane figure is one bounded by a line or lines drawn upon a plane.

The space contained within the boundary of a plane figure is called its surface; and its surface in reference to that of another figure, with which it is compared, is called its area.

The word figure, as here defined, is restricted to closed figures


according to the


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not be figures. The

word is, however,

very frequently B

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used in a wider sense to mean any combination of points, lines, or surfaces.

16. A circle is a plane figure contained by one (curved) line which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle.


Thus ABCDEFG is a circle, if all the straight lines which can be drawn from 0 to the circumference, such as OA, OB, OC, &c., are equal to one another; and O is the centre of the circle.

Strictly speaking, a circle is an inclosed space or surface, and the cir- B cumference is the line which incloses


Frequently, however, the word

circle is employed instead of circumfer


It is usual to denote a circle by three




letters placed at points on its circumference. The reason for this will appear later on.

17. A radius (plural, radii) of a circle is a straight line drawn from the centre to the circumference.

Thus OA, OB, OC, &c. are radii of the circle ACF.

18. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

Thus in the preceding figure BF is a diameter of the circle ACF.


19. Rectilineal figures are those which are contained by straight lines.

The straight lines are called sides, and the sum of all the sides is called the perimeter of the figure.

20. Rectilineal figures contained by three sides are called triangles.

21. Rectilineal figures contained by four sides are called quadrilaterals.

22. Rectilineal figures contained by more than four sides are called polygons.

Sometimes the word polygon is used to denote a rectilineal figure of any number of sides, the triangle and the quadrilateral being included.


First, according to their sides—

23. An equilateral triangle is one that

has three equal sides.

Thus, if AB, BC, CA are all equal, the triangle ABC is equilateral.


24. An isosceles triangle is one that has two equal sides.

Thus, if AB is equal to AC, the triangle ABC is isosceles.

25. A scalene triangle is one that has three unequal sides.

Thus, if AB, BC, CA are all unequal, the triangle ABC is scalene.



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Second, according to their angles

26. A right-angled triangle is one that has a right angle.

Thus, if ABC is a right angle, the triangle ABC is right-angled.


27. An obtuse-angled triangle is one that has an obtuse


Thus, if ABC is an obtuse angle, the

triangle ABC is obtuse-angled.



28. An acute-angled triangle is one that has three acute


Thus, if angles A, B, C are each of them acute, the triangle ABC is acute-angled.

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29. Any side of a triangle may be called the base. In an isosceles triangle, the side which is neither of the equal sides is usually called the base. In a right-angled triangle, one of the sides which contain the right angle is often called the base, and the other the perpendicular; the side opposite the right angle is called the hypotenuse.

Any of the angular points of a triangle may be called a vertex. If one of the sides of a triangle has been called the base, the angular point opposite that side is usually called the vertex.

Thus, if BC is called the base of a triangle ABC, A is the vertex.

30. If the sides of a triangle be prolonged both ways, nine angles are formed in addition to the angles of the triangle.

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