The corollary may be proved independently as follows: and this proof applies also to re-entrant polygons.

THEOREM 9, COR. The sum of the interior angles of a polygon of n sides is equal to (2n-4) right angles. Data ABCDE is a polygon of n sides.

To prove that the sum of the angles of a polygon is (2n-4) rt. S.

E

Construction Take any point O within the polygon.

Join O to all the vertices A, B, C....

Proof There are n triangles AOB, BOC, COD....

Fig. 11.

The sum of the angles of each triangle is 2 rt. ▲ s.

B

.. The sum of all the angles in the n triangles is 2n rt. ▲ s. But all the angles of the triangles make up the required angles of the polygon and in addition make up 4 rt. s round the point O, .. the required angles of the polygon = (2n − 4) rt. ≤ s.

Q. E. D. The above method of proof requires slight modification for some positions of the point O in certain cases of re-entrant polygons.

ANGLES AT A Point, ParaLLELS.

Ex. 6. From a point O in a straight line AOC, OB and OD are drawn on opposite sides of AC so that LAOB=LCOD (see fig. 12); prove that BOD is a straight line.

Ex. 7. Prove that the bisectors of vertically opposite angles are in the same straight line.

Ex. 8. If a straight line stands on another straight line, prove that the bisectors of the two adjacent angles so formed are at right angles to one another.

Ex. 9. A triangle ABC has

B

Fig. 12.

B=LC; D, E are points in AB, AC such that

DE is parallel to BC; prove that LADE=LAED.

Ex. 10. If a straight line is perpendicular to one of two parallel straight lines, it is also perpendicular to the other*.

Ex. 11. If each of two straight lines is perpendicular to a third straight line, the two straight lines are parallel to one another.

Ex. 12. If one angle of a parallelogram is a right angle, prove that all its angles must be right angles.

Ex. 18. AB, CD are two parallel straight lines drawn in the same sense, and P is any point inside the quadrilateral ABDC. Prove that

LBPD LABP+LCDP.

* This means that the truth of the statement is to be proved. Questions are often set in this form.

Ex. 14. The bisectors of two adjacent angles of a parallelogram are at right angles to one another.

Ex. 15. If a line is equally inclined to two intersecting lines, it is parallel to the bisector of one of the angles formed by the intersecting lines.

Ex. 16. Show how to draw a line through a point so as to be equally inclined to two intersecting lines.

ANGLES OF TRIANGLES*.

Ex. 17. A triangle has the two base angles equal. A side is produced beyond the vertex. Prove that the exterior angle is twice a base angle.

Ex. 18. Draw a triangle ABC; through A draw DAE parallel to BC. With this construction give another proof of Theorem 8.

Ex. 19. Find the sum of the angles of a pentagon by joining one vertex to the two opposite vertices.

Ex. 20. Prove that the sum of the angles of a polygon of n sides is (2n − 4) right angles by joining one vertex to each of the other vertices.

Ex. 21. BE and CF are the perpendiculars from B and C on to the opposite sides of the triangle ABC. Prove that LABE=LACF.

Ex. 22. ABC is a triangle and BE and CF the perpendiculars to the opposite sides meet in H. Prove that the angles BHC, BAC are equal or supplementary. Draw figures to show both cases.

Ex. 23. AOC is an acute angle and OB divides it into two unequal angles. From P, any point in OC, perpendiculars PN and PM are drawn to OA and OB. Prove that the angle between these perpendiculars is equal to the angle AOB.

Repeat the above for the case in which AOC is obtuse and each of the angles AOB, BOC is acute.

Ex. 24. If on the sides of an equilateral triangle three other equilateral triangles are described, show that the complete figure thus formed will be (i) a triangle, (ii) equilateral.

Ex. 25. If the bisector of an exterior angle of a triangle is parallel to one side, the triangle is isosceles.

Ex. 26. The perpendicular from the vertex to the hypotenuse in a rightangled triangle forms two triangles each equiangular with the original triangle. Ex. 27. The bisectors of the angles B, C of a triangle ABC intersect at I; prove that BIC=90° + LA.

Ex. 28. The bisectors of the exterior angles B and C of a triangle ABC intersect at l1; prove that BI1C=90° – }≤A.;

Ex. 29. From AB, the greatest side of a ▲ ABC, AD is cut off equal to AC. Prove that BCD =} (C – B).

Ex. 30. O is a point inside a triangle ABC; prove that BOC> <BAC. [Produce BO to cut AC.]

* Some of these Exx. assume the properties of the isosceles triangle.

Ex. 31. If a quadrilateral has pairs of adjacent angles equal, one pair of opposite sides are parallel; but, if opposite pairs of angles are equal, both pairs of opposite sides are parallel.

Ex. 32. Fig. 13 represents an equilateral triangle ABC with a line drawn from each vertex so as to make the marked angles equal. Prove that these lines enclose another equilateral triangle.

Ex. 33. From each vertex of any triangle draw lines to make equal angles with the sides as in fig. 13. Prove that these lines enclose a triangle which is equiangular with the given triangle.

Ex. 34. ABC is a triangle right-angled at C. BD=BC. The bisector of LA meets CD in E.

B

R

A

Fig. 13.

AB is produced to D so that
Prove LCEA is half a right

angle.

S. H. T. G.

2

CHAPTER III

CONGRUENT TRIANGLES, THE ISOSCELES TRIANGLE

NOTE ON THE METHOD OF SUPERPOSITION.

The method of proof used in Ths. 10 and 11 is called the "method of superposition." To test whether two figures are equal in all respects (congruent) we take up one, apply it to the other and see if it fits. This is a good practical method, but from a theoretical standpoint requires examination.

To test whether a line AB is equal to a line XY we apply AB to XY; if it fits we say that AB = XY; this in fact is the definition of equality. But how do we know that AB has not altered on the journey?

Or we might mark off the ends of AB on a straight-edge, and apply the straight-edge to XY. But how do we know that the straightedge has not altered? This brings us back to the former difficulty; a measuring instrument does not help us.

When we talk of moving a line, what we mean is a mark on paper, wood, etc.; an abstract line cannot be moved. We assume that the strip of paper does not alter in transit; in other words that, at any intermediate stage in the journey, it is equal to its former length. But how is this equality to be tested? By "application," as before. But this is arguing in a circle.

We shall not follow up this question further; enough has been said to show that the ultimate foundations of geometry involve very difficult questions. In fact, modern mathematicians have given up the attempt to prove Th. 10, and regard it as an unprovable assumption, or axiom. The theorem is given here as "traditional."

The symbol is used to denote "is congruent with."

THEOREM 10. If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles contained by those sides equal, the triangles are congruent.

Data ABC, DEF are two triangles which have AB = DE, AC = DF, and included A = included ▲ D.

Proof Apply AABC to ▲ DEF so that A falls on D, and AB falls along DE.

Since B falls on E and C on F, BC falls along EF; for there cannot be two different straight lines joining two points*.

.. AABC coincides with ADEF,

.. AABCE A DEF.

NOTE. The two congruent triangles might have been presented as in fig. 15: in this case it is necessary to turn one of them over before applying it to the other; compare figs. 17, 18, etc.

Q. E. D.

C F
Fig. 15.

*The same assumption as that which was made in Th. 4.