Elementary Plane Geometry: Inductive and Deductive.

Front matter

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Preface

(5) Many of the more difficult exercises at the end of each chapter have been simplified or omitted entirely. Those remaining have been carefully selected and graded. A number of new exercises have also been added.

(6) Mainly for the guidance of teachers, a summary has been placed at the end of each chapter in which are set forth all the important points dealt with in that chapter.

Thus, while we have endeavoured to preserve the spirit and characteristic features of Professor Baker's work, we trust that such alterations and additions as we have made will make this book more valuable as a text for use in the public schools.

Our thanks are specially due to Professor Baker who has left us perfectly free to make such alterations as we should deem

necessary to meet the conditions of our work.

•

J. C. PINCOCK A. C. CAMPBELL

Winnipeg, March 31, 1914

Contents

(At the close of Chapter VIII. the suggestion Is made that Chapters XVIII., XIX., and XX., relating to similar triangles, may at once he proceeded with.)

CHAPTER PAGE

I. Geometrical Elements 9

CHAPTER I. Geometrical Elements

A Straight Tine : __

It is evidently the shortest distance between its ends.

A broken line :

iwd. 1 t

A curved line : *i & cy'r;

An angle :

The size of the angle does not depend on the lengths of the bounding lines AB and AC, but on the amount of divergence of these lines from one another. Thus the angle P is greater than the angle Q, and the angle R is less than the angle Q.

It is. usual to indicate an angle by using one letter, as the angle P, or by using three letters, as the angle BAC. In the latter case the letter at the angle itself is in the middle, and the other two letters lie on the arms of the angle.

GEOMETRY.

If the line DB cuts the line AC at the point B, so as to make the angle DBA equal to the angle DBC, then each of these angles is called a right angle, and the lines DB and AC are said to be perpendicular to each other.

If the two angles on the other side of the line be measured*, they will be found to be equal, and hence are also right angles.

An angle which is less than a right . angle, as BAC, is called an acute, angle.

An angle which is greater than a right angle, as EDF, is called 3 an obtuse angle.

A circle is the usual figure described on a flat surface by means of the compasses.

Note the parts called centre, radius, and circumference.

All radii of the same circle are equal, since the ends of the compass legs remain the same distance apart while the circle is being described.

A line through the centre and terminated both ways by the circumference is called a diameter, as CD.

The part of the circle on each side of a diameter is called a semicircle.

GEOMETRICAL ELEMENTS.

A part of the circumference, as AB, is called an arc of the circle. The straight line joining A and B is called

a chord.

Any line drawn from a point without the circle and cutting it, is called a secant.

The circumference of any circle is supposed to be divided into 360 equal parts, each part being called a

degree.

If the arc AB contains 60 degrees, then the angle ACB at the centre is an angle of 60 degrees, expressed by 60°.

The lines AC, DE, through the centre, being perpendicular, each of the arcs AD, DC, CE, EA must contain 90°, and the angles ABD, DBC . . . are angles of 90°.

A semicircle contains 180°, and the straight angle ABC contains 180°.

A triangle :

It has three sides and three angles,,

A quadrangle:

It has four angles. Having four sides, it is also called a quadrilateral.

A straight line joining two opposite corners of a quadrilateral is called a diagonal.

GEOMETRY.

Figures contained by more than four straight lines are called polygons.

A straig-ht line has evidently throughout its entire length the same direction.

Two straight lines which have the same direction are said to be parallel to one another.

Parallel straight lines cannot intersect. For if they did, a' the point of intersection they would have different directions, and would therefore have different directions throughout their entire lengths, and hence would not be parallel.

To construct with the protractor at the point A in the line AB an angle of any required magnitude, say 63°: Place the centre of the

protractor at A, and let the line joining the centre with the point on the circumference which indicates 0°, rest along AB. At the point where the 63° line meets the circumference make a fine mark, C, on the paper. Removing the protractor, join AC. The angle BAG is of magnitude 63°.

Exercises.

411 figures in tills and succeeding exercises must be accurately constructed with instruments.

1. With the compasses describe circles having radii of lengths 5, 7, 10, . . . sixteenths of an inch.

2. With the protractor construct angles of magnitude 10°. 15°. 25°, 30° j 37°, 43% .... .

EXERCISES.

3. With the bevel construct a second set of angles of the foregoing magnitudes, using these angles to set the bevel.

4. Draw five straight lines of different lengths, and with the dividers and rule measure their lengths in inches and sixteenths of an inch.

5. Construct five angles, and, using the bevel, determine which is greatest and which least. Arrange them in order of magnitude. Using the protractor, measure their magnitude to the nearest degree.

6. Draw five straight lines of different lengths, and with the eye endeavor to judge their lengths in inches and fractions of an inch. Afterwards test the correctness of your judgment by actually measuring the lines.

7. Construct five angles of different magnitudes, and with the eye endeavor to judge the number of degrees in each. Afterwards test the correctness of your judgment by actually measuring the angles with the protractor.

8. With the eye endeavor to judge the lengths or heights of various objects in the room, at a distance from you. Afterwards test the correctness of your judgment by actually measuring the lengths or heights.

9. A and B being two distant objects and your eye being at C, endeavor with the eye to judge the angle which these objects subtend at your eye, i.e., the angle ACB. Afterwards sight the inside edges of the legs of the bevel towards A and B, and then placing the bevel on the protractor, roughly measure in this way the angle ACB, so correcting, if necessary, your judgment.

10. Draw any two lines of different lengths, and draw a line equal to their difference.

11. Draw any line, and draw another line three times as long as the former.

12. Construct two angles of different magnitudes, and with the bevel constructing two adjacent angles equal to them, form an angle equal to their difference. Measure with the protractor the number of degrees in the original angles and in the difference, and compare.

13. Construct two angles of different magnitudes, and with the bevel constructing two adjacent angles equal to them, form an angle equal to their sum. Measure with the protractor the number of degrees in the original angles and in the sum, and compare.

GEOMETRY.

14. Construct an angle of 30°. With the bevel construct two other angles equal to it, one on each side of the first, the three bounding lines radiating from the same point. What positions do the outside lines of your figure occupy with respect to each other, and why? Test with an instrument.

15. Construct an angle of 60°. With the bevel construct five other angles equal to it, each adjacent to the preceding, the bounding lines all radiating from the same point. What positions do the first and last lines of these angles occupy with respect to each ether, and why?

16. In the figure of the preceding question, if O be the point from which the lines radiate, measure off with the dividers on these lines equal lengths, OA, OB, OC, OD, OE, OF. What do you observe as to the lengths AB, BC, CD* DE, EF, FA ?

17. Fold a piece of paper so as to get a straight crease. Fold the crease over on itself. How many degrees in each ot the four angles so obtained, and why ?

18. With a needle mark two points. Join them, using ruler and a fine pencil. Turn the ruler over to the other side of the two points and again join them. What quality in the ruler may you test in this way ?

19. At points on your paper some distance from one another, construct two angles, as nearly as you can judge, equal. Test with an instrument the correctness of your judgment.

20. Through what angle does the minute-hand of a clock move in 20 minutes ? Through what angle does the hour-hand move in the saijne time ?

21. Describe a circle, and, supposing it intended for the face of a clock, mark the points where the usual Roman numerals should be placed.

22. One side of a piece of paper being a straight line, tear the remaining boundary into any irregular shape. With your protractor convert this paper into a protractor, so as to mark angles at intervals of 10°, the markings being on the irregular edge of the paper.

CHAPTER II. Construction of Triangles

1. Take a line AB of any length.

First with A as centre, then with B as centre, and in both cases with the same radius AB, describe portions of circles so that they intersect, as indicated, at C. Then the three lines A AB, BC, CA are all equal. The triangle CAB, which has thus all its sides equal, is called an equilateral triangle.

Adjust the bevel to each of the angles of this triangle, and compare their magnitudes.

Construct equilateral triangles whose sides are 14, 21, 30, 40 . . . sixteenths of an inch.

Apply the bevel to all the angles of these triangles, and compare their magnitudes.

Cut accurately any one of these equilateral triangles from the paper, and, clipping off the angles, fit them on one another, and on the angles of the other equilateral triangles, so as to compare their magnitudes.

The result of our observations is that the angles in an equilateral triangle are equal to one another, and are equal to the angles in any other equilateral triangle.

GEOMETRY.

2. Take a line AB of, say, 1 inch in length, and with centres A and B describe portions of circles intersecting as indicated at C, each circle having the same radius, say li inches. Draw lines from C to A and B. Then the triangle CAB has two sides equal.

A triangle with two of its sides equal is called an isosceles triangle.

Adjusting the bevel to the angles CAB and CBA, compare their magnitudes.

Compare also the sizes of these angles by accurately cutting the triangle out of the paper, and placing the triangle reversed in the vacant space left in the paper, so that the angle B rests in the space A.

Compare also the sizes of these angles by folding the triangle along the line from C to the middle of AB,

Construct the following isosceles triangles :

Base 1 in., each side 2 in.

Base 3 in., each side 2 in.

Base 2\ in., each side 2rf in.

In each case compare the magnitudes of the angles at the base.

The result of our observations is that the angles at the base of an isosceles triangle are equal.

Of course it would follow from this that all the angles in an equilateral triangle are equal, as we have already seen.

CONSTRUCTION OF TRIANGLES.

Prolong- the sides CA, CB, and adjusting the bevel to the angles BAD, ABE, on the other side of the base, they will be found to be equal. This may also be reasoned out as follows : The angles on one side of a straight line at any point in it make up 180°. But the angles CAB and CBA are equal.

Therefore the remaining angles BAD and ABE are also equal.

3. Taking any line AB, with the bevel or protractor construct equal angles at A and and produce the bounding lines of these angles to meet in C. Then employing the dividers or compasses, compare the lengths of the sides CA, CB, of the triangle CAB.

Construct the following triangles :

Base 1 in., each of angles at base 75°.

Base 3 in., each of angles at base 45°.

In each case compare the magnitudes of the sides adjacent to the equal angles.

The result of our observations is that if two angles of a triangle are equal, the sides opposite to these angles are also equal.

Of course it will follow that if the three angles of a triangle are equal, the three sides will also be equal, i.e., the triangle will be equilateral.

GEOMETRY.

4. Take a line AB of length li inches, and with centres A and B, and radii if and 2 inches respectively, describe portions of circles, so that they intersect at C. Join CA and CB.

We have thus a triangle CAB whose sides are unequal, called a scalene triangle.

With the bevel or protractor, compare the magnitudes of the three angles of this triangle.

What positions do the greatest, intermediate, and least angles occupy with regard to the greatest, intermediate, and least sides?

Construct the following triangles :

With sides 3, 5 and 6 inches.

With sides 3|, 4| and 2\ inches.

In each case measure the angles of the triangle drawn, and note their positions with regard to the greatest, intermediate, and least sides.

The result of our observations is that, the angles of a scalene triangle are unequal, and that the greatest, intermediate, and least angles are opposite respectively the greatest, intermediate and least sides.

This fact may also be stated as follows : —

If one side of a triangle be greater than another, the angle opposite the greater side is greater than the angle opposite the less.

CONSTRUCTION OF TRIANGLES.

At A in the line AB construct the angle BAD of 40° , and at B the angle ABC of 120°. Produce AD, BC to meet in E. Compare the length of the side EA with that of EB. Note which side has the greater angle opposite it.

Repeat the same experiment making : —

(1) The angle BAD 80° and the angle ABC 20°.

(2) The angle BAD 90° and the angle ABC 50°.

The result of? such observations will be found to be

that if one angle of a triangle be greater than another, the side opposite the greater angle is greater than the side opposite the less.

Verify this result by drawing several other triangles and examining as above.

5. Draw any equilateral triangle, any isosceles triangle, any scalene triangle. In the case of each of these triangles, lay off, with the bevel, three angles adjacent to one another, equal to the three angles of each triangle in the way indicated in the figure.

Determine the position of the first and last lines, LM and LK, with respect to each other.

The result of such an examination will be found to be that the lines KL and LM are in the same straight line, i.e., the sum of the three angles in each of these triangles is two right angles or 180°.

Verify this statement by drawing several triangles. Measure the three angles in each with the protractor, and find their sum.

GEOMETRY.

A good general proof of this statement is given below : —

Construct a triangle ABC, and place a pencil in the position DC.

Turn the pencil through the angle BCA, in the direction indicated by the arrow head, to the position EC. Slide it B along CA, towards A, to the position FG, and turn to the position HK. Slide

BL, and turn it through the angle B, to the position

BM.

The pencil has rotated through all the angles of the triangle. But in its final position BM it points in a direction just opposite to its first position DC, and therefore must have rotated through 180°. Hence all

the angles of this (which is any) triangle must together equal 180° , or two right angles.

What will be the magnitude of each angle of an equilateral triangle?

6- Draw any triangle ABC. Measure carefully the sides AB, AC, and BC. Add the lengths of AB and AC together and compare the result with BC. Add the lengths of AB and BC together and compare the result with AC. Add the lengths of AC and BC together and compare the result with AB.

Repeat the same experiment with several triangles and note your result. The result of such observations will be found to be that any two sides of a triangle are together greater than the third side.

A it through the angle CAB, it along AB to the position

EXERCISES.

Verify this statement by attempting- to draw triangles with sides of 2, 3 and 6 inches, 2, 4, and 6 inches.

Summary.

1. An .equilateral triangle is one which has its three sides equal. Its three angles are also equal, and each angle is equal to 60°.

2. An isosceles triangle is one which has two sides equal. The angles at the base of an isosceles triangle are equal, and if the equal sides be produced, the angles on the other side of the base will also be equal.

3. If two angles of a triangle be equal, the sides opposite these angles will also be equal.

4. A scalene triangle is one which has three unequal sides. The three angles of a scalene triangle are unequal, and the greatest, intermediate, and least angles are opposite respectively the greatest, intermediate, and least sides.

Or, if one side of a triangle be greater than another, the angle opposite the greater side is greater than the angle opposite the less.

Also, if one angle of a triangle be greater than another, the side opposite the greater angle is greater than the side opposite the less.

5. The sum of the three angles of any triangle is two right angles or 180°.

6. Any two sides of a triangle are together greater than the third side.

1. At a given point in a straight line, construct an angle of 60°, using only compasses and ruler.

2. Construct an isosceles triangle, and produce the base both ways. What do you note as to the magnitudes of the angles so formed outside the triangle ? Give reasons why this should be so.

3. The angle at the vertex of an isosceles triangle is 75°, and each of the equal sides is 2 inches. Construct the triangle. With a protractor, measure the remaining angles of this triangle. Verify your results by calculation.

4. The angle at the vertex of an isosceles triangle is 30°, and the base is two inches long. Construct the triangle, first calculating the magnitude of the angles at the base.

GEOMETRY.

5. Each of the angles at the base of an isosceles triangle is 72°, and the two sides opposite these angles are each 3 inches long. Construct the triangle.

6. Describe an equilateral triangle ABC. On BC describe an equi lateral triangle DBC ; on CA an equilateral triangle ECA ; and on AB an equilateral triangle FAB. What do you observe as to the positions of the lines DC, CE with respect to one another? What other lines show the same peculiarity ? Give reasons why this should be so.

7. In the preceding question, mark all the lines that are equal to one another. How many equilateral triangles are there in the figure ? Name them.

8. With centre A, outside a straight line, describe a circle of such radius as to cut the line in two points, B and C. What sort of triangle is ABC ?

9. In the figure of the preceding question find on the side of BC remote from A, a point D, such that a circle with D as centre can be described to pass through both B and C.

10. B and C being two points in a line, find on either side of the line points K, L; M, N, . . . such that a circle may be described, with any one of them as centre, to pass through B and C. What do you observe as to the positions of K, L, M, N, . . . with respect to one another?

11. Construct a scalene triangle ABC, and on the side of BC away from A, describe a triangle DBC, with DB = AB, and DC = AC. Join AD. What triangles in the figure are isosceles ? What inference can you draw as to the angles BAC and BDC?

CHAPTER III. Equality of Triangles

1. Construct two triangles, each with sides of lengths if, l\ and if inches, as indicated in the adjacent figures.

Adjust the bevel, or the protractor, to the angle A, and also to the angle D, and carefully compare the magnitudes of these angles. In like manner compare the magnitudes of the angles B and E, and also the magnitudes of the angles C and F.

Next cut both triangles from the paper, and place one triangle upon the other so that the corresponding angular points coincide. From this superposition what conclusion do you draw as to the areas of the triangles ?

Repeat the same construction, measurement, and superposition with two triangles whose sides are 4, 2 and 4^ inches.

GEOMETRY.

The result of our observations in these cases is that

if two triangles have their sides equal, the angles which are opposite to equal sides are equal and the areas are equal. In other words two such triangles are the same triangle in different positions.

Another way of stating the fact is to say that if the sides of a triangle are fixed, the angles are fixed, and the area is fixed.

2. Construct two angles, BAC and EDF, each of 30°. On sides of these angles measure off distances AB and DE, each of length l| inches; and also distances AC and DF, each of length 2 inches. Join BC and EF, thus forming two triangles, ABC and DEF.

Adjust the bevel, or protractor, to the angle B, and also to the angle E, and carefully compare the magnitudes of these two angles. In like manner compare the magnitudes of the angles C and F. With the dividers compare the magnitudes of the sides BC and EF. Further, cut one triangle from the paper, and place it

EQUALITY OF TRIANGLES.

upon the other. From this superposition what conclusion do you draw as to the areas of the two triangles?

Repeat the same construction, measurement, and superposition with the following triangles :

Two whose sides are if and 2f inches, and included angle 30°.

Two whose sides are if and 2 inches, and included angle 135°.

The result of our observations in all these cases is

that if two triangles have two sides in each equal, and the angles included by these two sides equal, then the remaining sides are equal, and the angles opposite to equal sides are equal, and the triangles are equal in area. In other words two such triangles are the same triangle in different positions.

Another way of stating the fact is to say that if two sides and the included angle of a triangle ^are fixed, the remaining side and angles are fixed, and the area is fixed.

3. Since, as we have shown in Chapter II, the three angles of any triangle together make 180°, therefore the sum of the three angles of any one triangle equals the sum of the three angles of any other triangle.

It follows that if two triangles have two angles in the one equal to two angles in the other, the third angle in one triangle is equal to the third angle in the other.

4. Construct two triangles, ABC, DEF, each with base if inch, and angles at the base 79° and 57°.

GEOMETRY.

It follows, from 4, that the remaining angles at A and D are equal, each being 44°. Putting the points of the dividers on A and B, and carrying the dividers, so adjusted, to DE, compare the magnitudes of AB and DE. In like manner compare the magnitudes of AC and DF.

A D

Next, cutting one of the triangles from *the paper, place it upon the other. From this superposition what conclusion do you draw as to the areas of the triangles ?

Repeat the same construction, measurement, and superposition with the following triangles :

Two whose bases are if in., and angles adjacent to base 38° and 110°.

Two whose bases are 3i in., and angles adjacent to the base 49° and 95°.

The result of our observations in all these cases is that if two triangles have their bases equal, and angles adjacent to the bases equal, the remaining angles are equal, and the sides

EQUALITY OF TRIANGLES.

opposite to equal angles are equal, and the areas are equal. In other words they are the same triangle in different positions.

Another way of stating the fact is to say that if a side of a triangle and the angles adjacent to this side are fixed, then the remaining angle and sides are fixed, and area is fixed.

5. It must be observed in connection with the results obtained in section 4, that if two triangles have one side and any two angles of one equal respectively to one side and the corresponding angles of the other, then the triangles will be equal in all respects ; for if two angles of one triangle are equal to two angles of the other, the remaining angles will also be equal. That is, we shall have one side and its adjacent angles in one triangle equal to one side and its adjacent angles in the other triangle.

E.g. , in the triangles ABC and DEF, suppose the side BC = side EF, the angle ABC = the angle DEF, and the angle BAG = the angle EDF.

It follows that the angle ACB = the angle DFE (sec. 3). Now, we know that BC = EF,

that the angle ABC = the angle DEF, and that the angle ACB = the angle DFE.

Therefore the triangles are equal in all respects.

6. Draw AC and DE any two straight lines cutting one another at B.

GEOMETRY.

The angles ABD and CBE are called vertically opposite angles, as also are the angles ABE and DBC.

Using the bevel, what do you a

observe with reference to the " ^

magnitudes of the vertically Opposite angles ABD and CBE ? 0

What with reference to the magnitudes of the angles

ABE, DBC ?

Draw other intersecting straight lines and note the magnitudes of vertically opposite angles.

We may demonstrate the relation between such angles as follows :

The angle ABD = 180° - the angle ABE.

The angle EBC also = 180° - the angle ABE.

. * . Angle ABD = angle EBC.

Show similarly that angle ABE = angle DBC.

Hence, if two straight lines cut one another, the vertically opposite angles are equal.

Summary.

1. By saying that two triangles are equal in all respects we mean that, the three sides of one triangle are respectively equal to the three sides of the other, the three angles of one triangle are respectively equal to the three angles of the other, and the triangles are equal in area.

2. Two triangles will be equal in all respects if : —

(1) Three sides of one are respectively equal to three sides of the other.

Or (2) Two sides and the contained angle of one are respectively equal to two sides and the contained angle of the other.

Or (3) One side and two angles of one are equal respectively ta one side and the corresponding angles of the other.

EXERCISES.

3. By saying that a triangle is fixed or determined we mean that from the information given, that triangle and no other may be drawn.

4. A triangle is fixed or determined if : —

(1) Three sides are given.

Or (2) Two sides and contained angle are given.

Or (3) Two angles and one side are given (the side being given either adjacent to the given angles or opposite one of them).

5. If two triangles have two angles of one respectively equal to two angles of the other, the third angle of the one is equal to the third angle of the other.

6. If two straight lines cut each other, the vertically opposite angles are equal.

. Exercises.

1. The sides of a triangle are 1J, 2 and 2£ inches. Construct the triangle ; and with the protractor measure the angles to the nearest degree.

2. Two sides of a triangle are 2£ and 1£ inches, and the included angle is 47°. Construct the triangle ; and measure the remaining side to the nearest sixteenth of an inch, and the remaining angles to the nearest degree.

3. Two sides of a triangle are and 6^ inches, and the included angle is 54°. Construct the triangle ; and measure the remaining side to the nearest sixteenth of an inch, and the remaining angles to the nearest degree.

4. Two angles of a triangle are 107° and 27°, and the side adjacent to them is 2 inches. Construct the triangle ; and measure the remaining angle to the nearest degree, and the remaining sides to the nearest sixteenth of an inch.