THEOREM 1. [Euc. I. 13.] The adjacent angles which one

straight line makes with another straight line on one side

of it are together equal to two right angles.

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COR. 1. If two straight lines cut one another, the four

angles so formed are together equal to four right angles.

COR. 2. When any number of straight lines meet at a

point, the sum of the consecutive angles so formed is equal

to four right angles.

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11

COR. 3. (i) Supplements of the same angle are equal.

(ii) Complements of the same angle are equal.

THEOREM 2. [Euc. I. 14.] If, at a point in a straight line,

two other straight lines, on opposite sides of it, make the

adjacent angles together equal to two right angles, then

these two straight lines are in one and the same straight line.

THEOREM 3. [Euc. I. 15.] If two straight lines cut one

another, the vertically opposite angles are equal.

Triangles.

DEFINITIONS

THE COMPARISON OF TWO TRIANGLES

THEOREM 4. [Euc. I. 4.] If two triangles have two sides

of the one equal to two sides of the other, each to each, and

the angles included by those sides equal, then the triangles

are equal in all respects.

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12

14

16

17

18

THEOREM 5. [Euc. I. 5.] The angles at the base of an isosceles

triangle are equal.

COR. 1. If the equal sides of an isosceles triangle are pro-

duced, the exterior angles at the base are equal.

COR. 2. If a triangle is equilateral, it is also equiangular.

THEOREM 6. [Euc. I. 6.] If two angles of a triangle are equal

to one another, then the sides which are opposite to the equal

angles are equal to one another.

THEOREM 7. [Euc. I. 8.] If two triangles have the three sides

of the one equal to the three sides of the other, each to each,

they are equal in all respects.

THEOREM 8. [Euc. I. 16.] If one side of a triangle is pro-

duced, then the exterior angle is greater than either of the

interior opposite angles.

COR. 1. Any two angles of a triangle are together less

than two right angles.

COR. 2.

angles.

COR. 3.

Every triangle must have at least two acute

Only one perpendicular can be drawn to a

straight line from a given point outside it.

THEOREM 9. [Euc. I. 18.] If one side of a triangle is greater

than another, then the angle opposite to the greater side is

greater than the angle opposite to the less.

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THEOREM 10. [Euc. I. 19.] If one angle of a triangle is

greater than another, then the side opposite to the greater

angle is greater than the side opposite to the less.

THEOREM 11. [Euc. I. 20.] Any two sides of a triangle are

together greater than the third side.

THEOREM 12. Of all straight lines from a given point to a

given straight line the perpendicular is the least.

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COR. 1. If OC is the shortest straight line from O to the

straight line AB, then OC is perpendicular to AB.

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COR. 2. Two obliques OP, OQ, which cut AB at equal

distances from C the foot of the perpendicular, are equal.

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COR. 3. Of two obliques OQ, OR, if OR cuts AB at the

greater distance from C the foot of the perpendicular, then

OR is greater than OQ.

Parallels.

PLAYFAIR'S AXIOM

THEOREM 13. [Euc. I. 27 and 28.] If a straight line cuts two

other straight lines so as to make (i) the alternate angles

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equal, or (ii) an exterior angle equal to the interior opposite

angle on the same side of the cutting line, or (iii) the interior

angles on the same side equal to two right angles; then in

each case the two straight lines are parallel.

THEOREM 14. [Euc. I. 29.] If a straight line cuts two parallel

lines, it makes (i) the alternate angles equal to one another;

(ii) the exterior angle equal to the interior opposite angle on

the same side of the cutting line; (iii) the two interior angles

on the same side together equal to two right angles.

PARALLELS ILLUSTRATED BY ROTATION.

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38

HYPOTHETICAL CON-

STRUCTION

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THEOREM 15. [Euc. I. 30.] Straight lines which are parallel

to the same straight line are parallel to one another.

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Triangles continued.

THEOREM 16. [Euc. I. 32.] The three angles of a triangle

are together equal to two right angles.

COR. 1. All the interior angles of any rectilineal figure,

together with four right angles, are equal to twice as many

right angles as the figure has sides.

COR. 2. If the sides of a rectilineal figure, which has no

re-entrant angle, are produced in order, then all the exterior

angles so formed are together equal to four right angles.

THEOREM 17. [Euc. I. 26.] If two triangles have two angles

of one equal to two angles of the other, each to each, and any

side of the first equal to the corresponding side of the other,

the triangles are equal in all respects.

ON THE IDENTICAL EQUALITY OF TRIANGLES

THEOREM 18. Two right-angled triangles which have their

hypotenuses equal, and one side of one equal to one side of

the other, are equal in all respects.

THEOREM 19. [Euc. I. 24.] If two triangles have two sides of

the one equal to two sides of the other, each to each, but the

angle included by the two sides of one greater than the angle

included by the corresponding sides of the other; then the

base of that which has the greater angle is greater than the

base of the other.

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2

44

46

48

50

51

52

THEOREM 20. [Euc. I. 33.] The straight lines which join the

extremities of two equal and parallel straight lines towards

the same parts are themselves equal and parallel.

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COR. 1. If one angle of a parallelogram is a right angle, all
its angles are right angles.

THEOREM 21. [Euc. I. 34.] The opposite sides and angles of a

parallelogram are equal to one another, and each diagonal

bisects the parallelogram.

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COR. 2. All the sides of a square are equal; and all its

angles are right angles.

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COR. 3. The diagonals of a parallelogram bisect one

another.

THEOREM 22. If there are three or more parallel straight lines,

and the intercepts made by them on any transversal are equal,

then the corresponding intercepts on any other transversal

are also equal.

COR. In a triangle ABC, if a set of lines Pp, Qq, Rr,...,

drawn parallel to the base, divide one side AB into equal parts,

they also divide the other side AC into equal parts.

DIAGONAL SCALES

Practical Geometry. Problems.

INTRODUCTION. NECESSARY INSTRUMENTS

PROBLEMS ON LINES AND ANGLES.

PROBLEM 1. To bisect a given angle.

PROBLEM 2. To bisect a given straight line.

PROBLEM 3. To draw a straight line perpendicular to a given

straight line at a given point in it.

PROBLEM 4. To draw a straight line perpendicular to a given

straight line from a given external point.

PROBLEM 5. At a given point in a given straight line to make

an angle equal to a given angle.

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76

PROBLEM 6. Through a given point to draw a straight line

parallel to a given straight line.

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PROBLEM 7. To divide a given straight line into any number

of equal parts.

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THE CONSTRUCTION OF TRIANGLES.

PROBLEM 8. To draw a triangle, having given the lengths of

the three sides.

PROBLEM 9. To construct a triangle having given two sides

and an angle opposite to one of them.

PROBLEM 10. To construct a right-angled triangle having given

the hypotenuse and one side.

THE CONSTRUCTION OF QUADRILATERALS.

PROBLEM 11. To construct a quadrilateral, given the lengths

of the four sides, and one angle.

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PROBLEM 12. To construct a parallelogram having given two

adjacent sides and the included angle.

PROBLEM 13. To construct a square on a given side.

Loci.

PROBLEM 14. To find the locus of a point P which moves so

that its distances from two fixed points A and B are always

equal to one another.

PROBLEM 15. To find the locus of a point P which moves so

that its perpendicular distances from two given straight lines

AB, CD are equal to one another.

INTERSECTION OF LOCI

THE CONCURRENCE OF STRAIGHT LINES IN A TRIANGLE.

I. The perpendiculars drawn to the sides of a triangle from

their middle points are concurrent.

II. The bisectors of the angles of a triangle are concurrent.

III. The medians of a triangle are concurrent.

COR.

The three medians of a triangle cut one another at a

point of trisection, the greater segment in each being towards

the angular point.

Areas.

PART II.

DEFINITIONS

99

THEOREM 23. AREA OF A RECTANGLE.

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THEOREM 24.

[Euc. I. 35.] Parallelograms on the same base

and between the same parallels are equal in area.

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AREA OF A PARALLELOGRAM

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105

THEOREM 25. AREA OF A TRIANGLE.

106

THEOREM 26. [Euc. I. 37.] Triangles on the same base and

between the same parallels (hence, of the same altitude) are

equal in area.

THEOREM 27. [Euc. I. 39.] If two triangles are equal in area,

and stand on the same base and on the same side of it, they

are between the same parallels.

THEOREM 28. AREA OF (i) A TRAPEZIUM.

(ii) ANY QUADRILATERAL.

AREA OF ANY RECTILINEAL FIGURE -

THEOREM 29. [Euc. I. 47. PYTHAGORAS'S THEOREM.] In a

right-angled triangle the square described on the hypotenuse

is equal to the sum of the squares described on the other two

sides.

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