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THEOREM 7. [Euclid 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.

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It is required to prove that the triangles are equal in all respects.

Proof.

Apply the

so that B falls on
so that A is on the

ABC to the ▲ DEF,

E, and BC along EF, and
side of EF opposite to D.

Then because BC EF, C must fall on F.

=

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.. the triangles are equal in all respects. Theor. 4.

Q.E.D.

Obs.

In this Theorem

it is given that AB-DE, BC= EF, and we prove that LC LF, LA=LD, Also the triangles are equal in area.

CA = FD;
LB=LE

Notice that the angles which are proved equal in the two triangles are opposite to sides which were given equal.

NOTE 1. We have taken the case in which DG falls within the L'EDF, EGF.

Two other cases might arise:

8

(i) DG might fall outside the 4a EDF, EGF [as in Fig. 1]. (ii) DG might coincide with DF, FG [as in Fig. 2].

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These cases will arise only when the given triangles are obtuseangled or right-angled; and (as will be seen hereafter) not even then, if we begin by choosing for superposition the greatest side of the ▲ ABC, as in the diagram of page 24.

NOTE 2. Two triangles are said to be equiangular to one another when the angles of one are respectively equal to the angles of the other. Hence if two triangles have the three sides of one severally equal to the three sides of the other, the triangles are equiangular to one another.

The student should state the converse theorem, and shew by a diagram that the converse is not necessarily true.

*

** At this stage Problems 1-5 and 8 [see page 70] may conveniently be taken, the proofs affording good illustrations of the Identical Equality of Two Triangles.

EXERCISES.

ON THE IDENTICAL EQUALITY OF TWO TRIANGLES.

THEOREMS 4 AND 7.

(Theoretical.)

1. Shew that the straight line which joins the vertex of an isosceles triangle to the middle point of the base,

(i) bisects the vertical angle:

(ii) is perpendicular to the base.

2. If ABCD is a rhombus, that is, an equilateral foursided figure; shew, by drawing the diagonal AC, that

(i) the angle ABC= the angle ADC;

(ii) AC bisects each of the angles BAD, BCD.

3. If in a quadrilateral ABCD the opposite sides are equal, namely ABCD and AD=CB; prove that the angle ADC=the angle ABC.

4. If ABC and DBC are two isosceles triangles drawn on the same base BC, prove (by means of Theorem 7) that the angle ABD=the angle ACD, taking (i) the case where the triangles are on the same side of BC, (ii) the case where they are on opposite sides of BC.

5. If ABC, DBC are two isosceles triangles drawn on opposite sides of the same base BC, and if AD be joined, prove that each of the angles BAC, BDC will be divided into two equal parts.

6.

Shew that the straight lines which join the extremities of the base of an isosceles triangle to the middle points of the opposite sides, are equal to one another.

7. Two given points in the base of an isosceles triangle are equidistant from the extremities of the base: shew that they are also equidistant from the vertex.

8. Shew that the triangle formed by joining the middle points of the sides of an equilateral triangle is also equilateral.

9. ABC is an isosceles triangle having AB equal to AC; and the angles at B and C are bisected by BO and CO: shew that

(i) BO=CO;

(ii) AO bisects the angle BAC.

10. Shew that the diagonals of a rhombus [see Ex. 2] bisect one another at right angles.

11. The equal sides BA, CA of an isosceles triangle BAC are produced beyond the vertex A to the points E and F, so that AE is equal to AF; and FB, EC are joined: shew that FB is equal to EC.

1.

EXERCISES ON TRIANGLES.

(Numerical and Graphical.)

Draw a triangle ABC, having given a=20′′, b=2·1′′, c=1·3′′. Measure the angles, and find their sum.

2. In the triangle ABC, a=7.5 cm., b=70 cm., and c=65 cm. Draw and measure the perpendicular from B on CA.

3. Draw a triangle ABC, in which a=7 cm., b=6 cm., C=65°. How would you prove theoretically that any two triangles having these parts are alike in size and shape? Invent some experimental illustration.

4.

Draw a triangle from the following data: b=2′′, c=2·5′′, A=57° ; and measure a, B, and C.

Draw a second triangle, using as B, and C; and measure b, c, and A.

data the values just found for a, What conclusion do you draw?

5. A ladder, whose foot is placed 12 feet from the base of a house, reaches to a window 35 feet above the ground. Draw a plan in which 1" represents 10 ft.; and find by measurement the length of the ladder. Plot my

6. I go due North 99 metres, then due East 20 metres. course (scale 1 cm. to 10 metres), and find by measurement as nearly as you can how far I am from my starting point.

7. When the sun is 42° above the horizon, a vertical pole casts a shadow 30 ft. long. Represent this on a diagram (scale 1" to 10 ft.); and find by measurement the approximate height of the pole.

8.

From a point A a surveyor goes 150 yards due East to B; then 300 yards due North to C; finally 450 yards due West to D. Plot his course (scale 1" to 100 yards); and find roughly how far D is from A. Measure the angle DAB, and say in what direction D bears from A.

9. B and C are two points, known to be 260 yards apart, on a straight shore. A is a vessel at anchor. The angles CBA, BCA are observed to be 33° and 81° respectively. Find graphically the approximate distance of the vessel from the points B and C, and from the nearest point on shore.

10. In surveying a park it is required to find the distance between two points A and B; but as a lake intervenes, a direct measurement cannot be made. The surveyor therefore takes a third point C, from which both A and B are accessible, and he finds CA=245 yards, CB=320 yards, and the angle ACB=42°. Ascertain from a plan the approximate distance between A and B.

THEOREM 8. [Euclid I. 16.]

If one side of a triangle is produced, then the exterior angle is greater than either of the interior opposite angles.

F

Let ABC be a triangle, and let BC be produced to D.

It is required to prove that the exterior LACD is greater than either of the interior opposite ABC, BAC.

Suppose E to be the middle point of AC

Join BE; and produce it to F, making EF equal to BE.

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and the AEB = the vertically opposite CEF; .. the triangles are equal in all respects;

Theor. 4.

so that the BAE the ECF.
But the ECD is greater than the
.. the ECD is greater than the
that is, the

ECF;

BAE ;

ACD is greater than the

BAC.

In the same way, if AC is produced to G, by supposing A to be joined to the middle point of BC, it may be proved that the 4 BCG is greater than the ABC.

But the BCG = the vertically opposite ACD.

.. the ACD is greater than the ABC.

Q.E.D.

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