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Examples I. Describe triangles whose sides have the following lengths :1. Base, 2"; left side, 2"; right side 1". 2. Base, 2"; left side, 1.5"; right side, 2.5". 3. Base, 3"; left side, 2"; right side, 1.5". 4. Base, 30 mm. ; left side, 70 mm.; right side, 50 mm. 5. Each of the three sides to measure 27 mm. 6. Base, 2.6"; left side, 1" ; right side, 2:4". 7. Base, 30 mm.; left side, 39 mm. ; right side, 60 mm.

8. It will be found impossible to draw a A if one of its sides is greater than the other two together. Try to draw a A whose sides measure 3, 1:5 and 1 inches; and another whose sides measure 20, 30, 60 mm,

2. On angles.-Any triangle has three corners. These corners are called angles. If the student examines the triangles which he has just drawn and the triangles on page 6 of this book, he will see that some of the corners are very sharp, so that if the triangle were made of metal these corners could be used for piercing. These sharp corners are small angles, and the more blunt corners are large angles. Thus, in the triangle given in Definition 28 on page 6, the angle at the right hand end of the base is small, and the angle at the left hand end is large.

Any two lines drawn from one point form an angle. Thus in Fig. 3 four angles are drawn of which the angle A is the smallest, the angle D the largest, and the angle C is larger than the angle B.

N
L
H

А к B M C P

R Fig. 3. When the two legs of a pair of compasses are opened they form an angle; and the wider we open them, the larger is the angle which they form. We could set the two legs of our compasses into each of the positions represented by the angles in Fig. 3. Read Definitions 8 and 9 on page 2.

The two lines which form an angle are called the arms of the angle, and the point where the two arms meet is called the vertex or sometimes the angular point. Thus in Fig. 3 the lines AH and AK are the two arms of the angle A, and the point A is the vertex of the angle or the angular point,

Up to the present we have been naming an angle by a letter placed at the angular point. This method is satisfactory so long as there is only one angle at that point; but in other cases it may lead to confusion. The correct way to name the first angle in Fig. 3 is the angle HAK;" notice the order of the letters; in the middle we place the angular point A, while the letters H and K are points one on each arm of the glė. It does not matter whether we place H or k first or last so long as we keep A in the middle; so that we may call the angle KAHif we choose. Similarly the fourth angle in Fig. 3 may be called either QDR or RDQ. In Fig. 2, the angle at the left hand end of the base should be called the angle BAC (or CAB), and the angle at the top of the triangle should be called the angle ACB (or BCA).

3. On equality of angles.-It is very important to notice that the size of an angle does not depend on the lengths of its arms but merely on the shape at the corner. The size of an angle depends only on the difference in the directions of its arms. In Fig. 3 if we lengthen or shorten one or both of the arms of the angle HAK we should not alter the size of the angle, so long as we do not alter the shape at the corner A.

Two angles are equal if we can place one angle on the other in such a way that the corners fit. Thus the angle at Q in the triangle PQR [Fig. 4) is equal to the angle at Y in the triangle XYZ; for we can place the angle at y on to the angle at Q in Р

Р

Р
X Х.

Х

R Y Z
QY Z

Y RZ
Fig. 4.

Fig. 5.

Fig. 6. such a way that the corners at Y and Q exactly fit (as shown in Fig. 5). But the angle at R is smaller than the angle at Z (Fig. 4) for if we place the corner Z on to the corner R we obtain Fig. 6, from which it is evident that the arms RQ and RP are less wide apart than the arms ZY and ZX.

Exam II.

(For answers see page xxiv.) Turn to page 14, and look at the triangle DEF;1. Name the angle at the vertex of the triangle. 2. Name the angle on the left side of the triangle.

Turn to the triangles in Fig. 4:-
3. What are the arms of the angle whose vertex is P?
4. What are the arms of the angle whose vertex is Y ?
5. Name the angle whose arms are QP and QR.
6. Name the angle whose arms are ZY and YX.

7. Which do you consider the smallest angle (i.e. the “sharpest corner”) in the triangle PQR ?

8. Which do you consider the largest angle in the triangle PQR ? 9. Which do you consider the largest angle in the triangle XYZ ? 10. Is the angle at P greater or less than the angle at X ?

K

H

are

B

4. Now consider Fig. 7. We have eleven different lines drawn from P, and they form ten small equal angles, viz. the angles A PB, BPC, CPD,

E DPE, etc. The arms of these angles

F of different lengths, but the

M angles themselves have been drawn equal. Any two of the lines which meet

A at p form an angle, for example the lines PC and PE form an angle CPE.

Fig. 7. Also it is obvious that this angle CPE is made up of two of the small equal angles, viz. CPD and DPE. Thus this angle CPE is twice as large as the angle CPD or twice as large as the angle APB.

In the same way the angle EPH is made up of three of the small equal angles, -viz. EPF, FPG, and GPH. Hence the angle EPH is three times as large as the angle APB.

We shall frequently use the symbol < for angle.

Examples III. Using Fig. 7:

1. How many times is 2 APB contained in each of the following angles ? – (a) DPF, (6) FPL, (c) APG, (d) MPA.

2. Name the line which divides LKPE into two equal parts. 3. Name the line which divides LKPC into two equal parts.

4. PM is one arm of an angle which is five times as great as LAPB; which is the other arm?

5. How many times is 2 BPD contained in <LPF?

5. On Right Angles. In Fig. 8 the line CD is said to “stand upon” the line AB. In such a figure there are two angles formed at C, viz. the angles BCD and ACD. In Fig. 8 LACD is obviously larger than ZBCD; but in Fig. 9 where

the line RS stands on the line PQ, the two angles PRS and QRS are equal. When one line stand

D

is ing upon another line forms two equal angles each of these angles Ā

с B P R is called a right angle. Thus each of the angles

Fig. 8.

Fig. 9. PRS and QRS is a right angle. The lines PQ and RS are said to be perpendicular to each other, or at right angles to each other.

We shall frequently use the symbol I for perpendicular,

Right angles are very common and are easily recognised by the eye. In Fig. 10 ABCD represents an ordinary sheet of paper; each of the angles A, B, C, D is

B a right angle. Any angle is a right angle if we can fit the corner of an ordinary sheet of paper on to it; for example the angles of the cover of a book, the angles of an ordinary picture frame, the angles of a box-lid, etc. In Fig. 10, AB is perpendicular to BC, AD is perpendicular to DC, and so on.

Fig. 10. Your two “set squares are wooden triangles, and in each of these one angle is obviously a right angle. Any triangle which has one angle a right angle is called a rightangled triangle.

Now learn Definitions 11 and 12 on page 4. Note that the words obtuse and acute are Latin words which mean blunt and sharp. Learn also Definitions 24-29

A

D

on

page 6.

A

D

C

B

6. The following problems should now be practised till the student can do them from memory :

Fig. 11. Problem II.- At a given point C in a given straight line AB,

to draw a line perpendicular to AB. (See Fig. 11.)

From C mark off equal portions CD and of any convenient length. With centres D and E, and any convenient equal radii, describe arcs of circles intersecting at F. Join CF.

Then CF is perpendicular to AB.

NOTE 1.-If the point C is very near to one end of the line it will be necessary to produce the line (i.e. lengthen it by means of the ruler) before we start the construction given above.

NOTE 2.–The perpendicular CF can be more easily (though less neatly) drawn by placing the right-angled corner of a set square at C, with one edge lying along CA or CB. We can then rule CF along the other edge.

Problem III.From a given point C without a given straight line AB, to draw a line perpendicular to AB. (See Fig. 12.) With centre C and any

convenient radius draw arcs cutting AB in D and E. With centres D and E and

с any convenient equal radii describe

A D

E

B arcs cutting one another in F (on the other side of the line from C). Join CF cutting AB at G. Then CG is perpendicular to AB.

Fig. 12. NOTE 1.--Here again it may be necessary to produce AB, before the above construction is possible.

NOTE 2.-We can place a set square so that one arm of the right angle lies along AB, and the other passes through C, and then rule CG.

NOTE 3.—The point G is called the foot of the perpendicular CG.

Problem IV.-Given a line, AB, to bisect it, or divide it into two equal parts. (See Fig. 13.)

With A as centre, and radius rather greater than half AB, describe an arc. With B as centre and an equal radius, describe

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a second arc, intersecting the first one in points C and D. Join CD, cutting AB in P, the required point.

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