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EXERCISES ON INEQUALITIES IN A TRIANGLE.

1. The hypotenuse is the greatest side of a right-angled triangle.

2.

The greatest side of any triangle makes acute angles with each of he other sides.

3. If from the ends of a side of a triangle, two straight lines are drawn to a point within the triangle, then these straight lines are together less than the other two sides of the triangle.

4. 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.

5. 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.

6. 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.

7. 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.

8. The difference of any two sides of a triangle is less than the third side.

9. The sum of the distances of any point from the three angular points of a triangle is greater than half its perimeter.

X. 10. The perimeter of a quadrilateral is greater than the sum of its diagonals.

11. ABC is a triangle, and the vertical angle BAC is bisected by a line which meets BC in X; shew that BA is greater than BX, and CA greater than CX. Hence obtain a proof of Theorem 11.

12.

The sum of the distances of any point within a triangle from its angular points is less than the perimeter of the triangle.

13. The sum of the diagonals of a quadrilateral is less than the sum of the four straight lines drawn from the angular points to any given point. Prove this, and point out the exceptional case.

14. In a triangle any two sides are together greater than twice the median, which bisects the remaining side.

[Produce the median, and complete the construction after the manner of Theorem 8.]

15. In any triangle the sum of the medians is less than the perimeter.

PARALLELS.

DEFINITION. Parallel straight lines are such as, being in the same plane, do not meet however far they are produced beyond both ends.

NOTE. Parallel lines must be in the same plane. For instance, two straight lines, one of which is drawn on a table and the other on the floor would never meet if produced; but they are not for that reason necessarily parallel.

AXIOM. Two intersecting straight lines cannot both be parallel to a third straight line.

In other words:

Through a given point there can be only one straight line parallel to a given straight line.

This assumption is known as Playfair's Axiom.

DEFINITION. When two straight lines AB, CD are met by a third straight line EF, eight angles are formed, to which for the sake of distinction particular names are given.

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of EF. Such angles are also known as corresponding angles. Similarly 7 and 3, 8 and 4, 1 and 5 are pairs of corresponding angles.

THEOREM 13. [Euclid I. 27 and 28.]

If a straight line cuts two other straight lines so as to make

(i) the alternate angles 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.

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(i) Let the straight line EGHF cut the two straight lines AB, CD at G and H so as to make the alternate 4 AGH, GHD equal to one another.

It is required to prove that AB and CD are parallel.

Proof. If AB and CD are not parallel, they will meet, if produced, either towards B and D, or towards A and C.

If possible, let AB and CD, when produced, meet towards B and D, at the point K.

Then KGH is a triangle, of which one side KG is produced to A; .. the exterior LAGH is greater than the interior opposite GHK; but, by hypothesis, it is not greater.

.. AB and CD cannot meet when produced towards B and D. Similarly it may be shewn that they cannot meet towards A and C:

.. AB and CD are parallel.

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(ii) Let the exterior EGB = the interior opposite GHD. It is required to prove that AB and CD are parallel.

Proof.

Because the LEGB = the GHD,

and the EGB = the vertically opposite AGH;
.. the LAGH = the GHD:

and these are alternate angles;

.. AB and CD are parallel.

(iii) Let the two interior BGH, GHD be together equal to two right angles.

It is required to prove that AB and CD are parallel.

Proof.

angles;

Because the BGH, GHD together two right

and because the adjacent

angles;

..the

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BGH, AGH together = two right

BGH, AGH together the 4o BGH, GHD.

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For instance, in the above diagram the line EGHF, which crosses the given lines AB, CD is a transversal.

THEOREM 14. [Euclid 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.

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Let the straight lines AB, CD be parallel, and let the straight line EGHF cut them.

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(ii) the exterior LEGB = the interior opposite L GHD ;

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(iii) the two interior L BGH, GHD together two right angles.

Proof. (i) If the AGH is not equal to the GHD, suppose the PGH equal to the GHD, and alternate to it; then PG and CD are parallel.

But, by hypothesis, AB and CD are parallel;

Theor. 13.

..the two intersecting straight lines AG, PG are both parallel to CD which is impossible.

Playfair's Axiom.

L

..the AGH is not unequal to the GHD;
that is, the alternate LAGH, GHD are equal.

(ii) Again, because the LEGB = the vertically opposite

LAGH;

and the AGH = the alternate

GHD;

Proved.

.. the exterior LEGB = the interior opposite / GHD.

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