Plane and Solid Geometry

157. The Indirect Method of Proof. The method of proof that assumes the proposition false and then shows that this assumption is absurd is called the indirect method or the reductio ad absurdum.

This method forms a kind of last resort in the proof of a proposition, after the synthetic and analytic methods have failed.

EXERCISE 18

1. Given ABC and ABD, two triangles on the same base AB, and on the same side of it, the vertex of each triangle being outside the other triangle. Prove that if AC equals

AD, then BC cannot equal BD.

Assume that BC = BD and show that the result is absurd, since it would make D fall on C, which is contrary to the given conditions.

2. On the sides of the angle XOY two equal segments OA and OB are taken. On AB a triangle APB is constructed with AP greater than BP.

Prove that OP cannot

bisect the angle XOY.

Assume that OP does bisect XOY. What is

the result? Is this result possible?

3. From M, the mid-point of a line AB, MC is drawn oblique

to AB. Prove that CA cannot equal CB.

Assume that CA does equal CB. What is the result? Is this result possible?

4. If perpendiculars are drawn to the sides.

of an acute angle from a point within the angle, they cannot inclose a right angle or an acute angle.

Assume that they inclose a right angle and show that this leads to an absurdity. Similarly for an acute angle.

5. One of the equal angles of an isosceles triangle is five ninths of a right angle. Prove that the angle at the vertex cannot be a right angle.

Assume that it is a right angle. Is the result possible ?

158. General Suggestions for proving Theorems. The following general suggestions will often be helpful:

1. Draw the figures as accurately as possible.

This is especially helpful at first. A proof is often rendered difficult simply because the figure is carelessly drawn. If one line is to be laid off equal to another, or if one angle is to be made equal to another, do this by the help of the compasses or by measuring with a ruler.

2. Draw as general figures as possible.

If you wish to prove a proposition about a triangle, take a scalene triangle. If an equilateral triangle, for example, is taken, it may lead to believing something true for every kind of a triangle, when, in fact, it is true for only that particular kind.

3. After drawing the figure state very clearly exactly what is given and exactly what is to be proved.

Many of the difficulties of geometry come from failing to keep in mind exactly what is given and exactly what is to be proved.

4. Then proceed synthetically with the proof if you see how to begin. If you do not see how to begin, try the analytic method, stating clearly that you could prove this if you could prove that, and so on until you reach a known proposition.

5. If two lines are to be proved equal, try to prove them corresponding sides of congruent triangles, or sides of an isosceles triangle, or opposite sides of a parallelogram, or segments between parallels that cut equal segments from another transversal.

6. If two angles are to be proved equal, try to prove them alternate-interior or exterior-interior angles of parallel lines, or corresponding angles of congruent triangles, or base ångles of an isosceles triangle, or opposite angles of a parallelogram.

7. If one angle is to be proved greater than another, it is probably an exterior angle of a triangle, or an angle opposite the greater side of a triangle.

8. If one line is to be proved greater than another, it is prob ably opposite the greater angle of a triangle.

EXERCISE 19

Prove the following propositions referring to equal lines:

1. If the sides AB and AD of a quadrilateral ABCD are equal, and if the diagonal AC bisects the angle at A, then BC is equal to DC.

2. A line is drawn terminated by two parallel lines. Through its mid-point any line is drawn terminated by the parallels. Prove that the second line is bisected by the first.

3. In a parallelogram ABCD the line BQ bisects AD, and DP bisects BC. Prove that BQ and DP trisect AC.

4. On the base AB of a triangle ABC any point P is taken. The lines AP, PB, BC, and CA are bisected by W, X, Y, and Z respectively. Prove that XY is equal to WZ.

5. In an isosceles triangle the medians drawn to the equal sides are equal.

6. In the square ABCD, CD is bisected by Q, and P and R are taken on AB so that AP equals BR. Prove that PQ equals RQ.

7. In this figure AC = BC, and AP BQ =

CR CS. Prove that QR = PS.

8. From the vertex and the mid-points of the equal sides of an isosceles triangle lines are drawn perpendicular to the base. Prove that they divide the base into four equal parts.

9. In the quadrilateral ABCD it is known that AB is parallel to DC, and that angle C equals angle D. On CD two points are taken such that CP=DQ. Prove that AP=BQ.

D Q

EXERCISE 20

Prove the following propositions referring to equal angles : 1. In this figure it is given that AC=BC, and that BQ and AR bisect the angles YBC and CAX respectively. Prove that ▲ APB is isosceles.

2. If through the vertices of an isosceles triangle lines are drawn parallel to the opposite sides, they form an isosceles triangle.

3. If the vertical angles of two isosceles triangles coincide, the bases either coincide or are parallel.

4. In which direction must the side of a triangle be produced so as to intersect the bisector of the opposite exterior angle?

Consider the cases, ▲ A<ZC, ZA=2C, 2A><C.

5. The bisectors of the equal angles of an isosceles triangle form, together with the base, an isosceles triangle.

6. The bisectors of the base angles of an equilateral triangle form an angle equal to the exterior angle at the

vertex of the triangle.

7. If the bisector of an exterior angle of a triangle is parallel to the opposite side, the triangle is isosceles.

8. A line drawn parallel to the base of an isosceles triangle makes equal angles with the sides or the sides produced.

9. A line drawn at right angles to AB, the base of an isosceles triangle ABC, cuts AC at P and BC produced at Q. Prove that PCQ is an isosceles triangle.

10. In this figure, if AB = CD, and A=LC, then BD is parallel to AC.

EXERCISE 21

Prove the following propositions by showing that two triangles are congruent :

1. A perpendicular to the bisector of an angle forms with the sides an isosceles triangle.

2. If two lines bisect each other at right angles, any point in either is equidistant from the extremities of the other.

3. From B a perpendicular is drawn to the bisector of the angle A of the triangle ABC, meeting it at X, and meeting AC or AC produced at Y. Prove that BX=XY.

4. If through any point equally distant from two parallel lines two lines are drawn cutting the parallels, they intercept equal segments on these parallels.

5. If from the point where the bisector of an angle of a triangle meets the opposite side, parallels are drawn to the other two sides, and terminated by the sides, these parallels are equal.

6. The diagonals of a square are perpendicular to each other and bisect the angles of the square.

7. If from a vertex of a square there are drawn line-segments to the mid-points of the two sides not adjacent to the vertex, these line-segments are equal.

8. If either diagonal of a parallelogram bisects one of the angles, the sides of the parallelogram are

all equal.

9. On the sides of any triangle ABC equilateral triangles BPC, CQA, ARB are constructed. Prove that AP=BQ=CR.

How can we prove that ▲ ABP is congruent to ARBC? Also that ▲ ARC is congruent to AABQ? Does this prove the proposition?

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