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PROPOSITION 19. THEOREM.
If one angle of a triangle be greater than another, then the side opposite to the greater angle shall be greater than the side opposite to the less.
Let ABC be a triangle in which the angle ABC is greater than the angle ACB.
Then shall the side AC be greater than the side AB.
But AC is not equal to AB, for then the angle ABC would be equal to the angle ACB; 1. 5. but it is not.
Нур. . Neither is AC less than AB; for then the angle ABC would be less than the angle ACB; 1. 18. but it is not.
Нур. . That is, AC is neither equal to, nor less than AB.
Therefore AC is greater than AB. Q.E.D.
NOTE. The mode of demonstration used in this Proposition is known as the Proof by Exhaustion. It is applicable to cases in which one of certain suppositions must necessarily be true ; and it consists in shewing that each of these suppositions is false with one exception : hence the truth of the remaining supposition is inferred.
Euclid enunciated Proposition 19 as follows :
The greater angle of every triangle is subtended by the greater side, or, has the greater side opposite to it.
[For Exercises on Props. 18 and 19 see page 44.]
Any two sides of a triangle are together greater than the third side.
Let ABC be a triangle. Then shall any two of its sides be together greater than the third side :
namely, BA, AC, shall be greater than CB;
AC, CB shall be greater than BA;
and CB, BA shall be greater than AC. Construction. Produce BA to D, making AD equal to AC. I. 3.
Then in the triangle ADC,
Constr. therefore the angle ACD is equal to the angle ADC. 1. 5. But the angle BCD is greater than its part the angle ACD; therefore also the angle BCD is greater than the angle ADC,
that is, than the angle BDC.
And in the triangle BCD, because the angle BCD is greater than the angle BDC, therefore the side BD is greater than the side CB.
that AC, CB are together greater than BA;
[For Exercises see page 44.]
PROPOSITION 21. THEOREM. If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, then these straight lines shall be less than the other two sides of the triangle, but shall contain a greater angle.
Let ABC be a triangle, and from B, C, the ends of the side BC, let the straight lines BD, CD be drawn to a point D within the triangle Then (i) BD and DC shall be together less than BA and AC;
(ii) the angle BDC shall be greater than the angle BAC. Construction. Produce BD to meet AC in E. Proof. (i) In the triangle BAE, the two sides BA, AE are together greater than the third side BE;
1. 20. to each of these add EC; then BA, AC are together greater than BE, EC. Ax. 4. Again, in the triangle DEC, the two sides DE, EC are together greater than DC;
I. 20. to each of these add BD; then BE, EC are together greater than BD, DC. But it has been shewn that BA, AC are together greater than BE, EC:
still more then are BA, AC greater than BD, DC. (ii) Again, the exterior angle BDC of the triangle DEC is greater than the interior opposite angle DEC;
I. 16. and the exterior angle DEC of the triangle BAE is greater
than the interior opposite angle BAE, that is, than the
angle BAC; still more then is the angle BDC greater than the angle BAC.
I. 16. EXERCISES.
ON PROPOSITIONS 18 AND 19.
1. The hypotenuse is the greatest side of a right-angled triangle.
2. If two angles of a triangle are equal to one another, the sides also, which subtend the equal angles, are equal to one another. Prove this [i.e. Prop. 6] indirectly by using the result of Prop. 18.
3. BC, the base of an isosceles triangle ABC, is produced to any point D; shew that AD is greater than either of the equal sides.
4. If in a quadrilateral the greatest and least sides are opposite to one another, then each of the angles adjacent to the least side is greater than its opposite angle.
5. In a triangle ABC, if AC is not greater than AB, shew that any straight line drawn through the vertex A and terminated by the base BC, is less than AB.
6. ABC is a triangle, in which OB, OC bisect the angles ABC, ACB respectively: shew that, if AB is greater than AC, then OB is greater than OC.
ON PROPOSITION 20.
7. The difference of any two sides of a triangle is less than the third side.
8. In a quadrilateral, if two opposite sides which are not parallel are produced to meet one another; shew that the perimeter of the greater of the two triangles so formed is greater than the perimeter of the quadrilateral.
9. The sum of the distances of any point from the three angular points of a triangle is greater than half its perimeter.
10. The perimeter of a quadrilateral is greater than the sum of its diagonals.
11. Obtain a proof of Proposition 20 by bisecting an angle by a straight line which meets the opposite side.
ON PROPOSITION 21. 12. In Proposition 21 shew that the angle BDC is greater than the angle BAC by joining AD, and producing it towards the base.
13. sum of the distances of any point within a triangle from its angular points is less than the perimeter of the triangle.
QUESTIONS FOR REVISION.
1. Define the complement of an angle. When are two angles said to be supplementary? Shew that two angles which are supple. mentary to the same angle are equal to one another.
2. What is meant by an angle being bisected internally and externally ?
Prove that the internal and external bisectors of an angle are at right angles to one another.
3. Prove that the sum of the angles formed by any number of straight lines drawn from a point is equal to four right angles.
4. Why must every triangle have at least two acute angles ? Quote. the enunciation of the proposition from which this inference is drawn.
5. In the enunciation The greater side of a triangle has the greater angle opposite to it, point out what is assumed and what is to be proved.
6. What is meant by the Proof by Exhaustion ? Illustrate the use of this method by naming the steps in the proof of Proposition 19.
7. What inference may be drawn respecting the triangles whose sides measure
(i) 4 inches, 5 inches, 4 inches ;
(iii) 6 inches, 10 inches, 4 inches ? 8. Quote the enunciations of propositions which, from a hypothesis relating to the sides of triangle, establish a conclusion relating to the angles.
9. Quote the enunciations of propositions which, from a hypothesis relating to the angles of a triangle, establish a conclusion relating to the sides.
10. Explain why parallel straight lines must be in the same plane.
11. Prove by means of Prop. 7 that on a given base and on the same side of it only one equilateral triangle can be drawn.
12. In an isosceles triangle, if the equal sides are produced, shew that the angles on the other side of the base must be obtuse.