4. O is any point within the triangle ABC; prove that OA+ OB + OC are less than the sum and greater than half the sum of AB + BC + CA.

5. Prove that the sum of the four sides of a quadrilateral figure is greater than the sum and less than twice the sum of the diagonals.

6. If ABC is a triangle in which AB is greater than AC, and D is the middle point of BC, and AB is joined, prove that the angle ADB is an obtuse angle.

7. Prove that the sum of the three sides of a triangle is greater than the sum of the three medians.

NOTE. The median of a triangle is the line that joins any angle to the middle point of the opposite side.

8. Prove that the sum of the three medians of a triangle is greater than half the sum of the sides.


1. Give the meaning and derivation of the words triangle, peria meter, isosceles, equilateral, hypotenuse, median.

2. If a triangle is isosceles, the angles at its base will be equal.
Enunciate the obverse, converse and contrapositive theorems.
3. Apply Theorem 7 to find the height of a tower.
4. Prove Theorem 6 in the manner of Theorem 8.

5. Why cannot Theorem 15 be proved in the same manner as Theorem 5?

6. Prove that only one perpendicular can be drawn from a given point to a given straight line.

7. Prove fully the corollary to Theorem 19.

8. Enumerate the five cases in which the equality of three parts in a pair of triangles involves the equality in all respects.

9. Mention cases, and draw the figures, in which two triangles are equal in three respects but not in all.

10. Prove fully the corollaries to Theorem 20.

1. Prove Theorem 19 by conceiving the figure to be folded down over the line AB, O falling on a point O', and RO, QO, PO, on RO', QO', PO', and using Theorems 12 and 13.

12. In Theorem 9, prove fully that ACD is greater than ABC.

13. Why is it necessary, in the enunciation of Theorem 9, to say interior and opposite angles?

14. What is the magnitude indirectly measured in Theorem 9?

15. Enunciate Theorem 10 formally. Is it merely the obverse of Theorem 6, or does it contain an additional geometrical fact ?

16. Prove Theorem 10 by reversal and superposition, using Theorem 9.

17. Shew how Theorem 12 depends ultimately on the Axioms.

18. Which Theorem in this Section proves that as you increase the angle between the legs of a pair of compasses you also increase the distance between their points ?

19. Shew the relation of Theorems 14, 15 and 16 to Theorem 5. 20. Enunciate the contra-positives of Theorems 9 and 18.



Def. 35. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet.

Axiom 5. Two straight lines that intersect one another cannot both be parallel to the same straight line.

Def. 36. A trapezium is a quadrilateral that has only one pair of opposite sides parallel.

This figure is sometimes called a trapezoid.

Def. 37. A parallelogram is a quadrilateral whose opposite sides are parallel.

Def. 38. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to one another.

Thus in the figure 1, 2, 7, 8 are called exterior angles, and 3, 4, 5, 6, interior angles; again, 4 and 6, 3 and 5, are called alternate angles; lastly, I and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles.

Def. 39. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted between perpendiculars let fall on it from the extremities of the former.

Thus the projections of AB, CD on EF are the lines ab, cd respectively.


It is clear that the line EF must be supposed indefinitely long. There could be no projection of AB on the terminated line GF.


If one straight line intersects two other straight lines so as to make the alternate angles equal, the straight lines are parallel.

Part. En. Let ABCD intersect EF and GH, and make the angle EBC equal to its alternate angle BCH; it is required to prove that EF is parallel to GH.

Proof. For if EF and GH meet towards F, H, they would form a triangle with BC;

and EBC would be its exterior angle, and therefore greater than the interior and opposite angle BCH. (Th. 9.) But EBC is equal to BCH,

(Hyp.) therefore EF and GH do not meet towards F, H.

Similarly they do not meet towards E, G; that is, EF is parallel to GF*.

(Def. 35.) Q. E. D.

THEOREM 22. If two straight lines are parallel, and are intersected by a third straight line, the alternate angles are equalt.

Part. En. Let EF and GH be parallel straight lines, and let ABCD intersect them; it is required to prove that the alternate angles EBC, BCH are equal.

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Proof. For if EBC were not equal to BCH, let some other line LBM be drawn through B making the angle LBC equal to the alternate angle BCH; then LM would be parallel to GH.

(Th. 21.) But EF is parallel to GH;

(Hyp.) that is, two intersecting lines LM, EF would be both parallel to GH; which is impossible.

(Ax. 5.) Therefore EBC is equal to BCH, that is, the alternate angles are equal.

Q. E. D. * Euclid, I. 27. + Euclid, 1. 29.

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