Obs. In this Theorem we must carefully observe what is given and what is proved.

From these data we prove that the triangles coincide on superposition.

also that the triangles are equal in area.

Notice that the angles which are proved equal in the two triangles are opposite to sides which were given equal.

NOTE. The adjoining diagram shews that in order to make two congruent triangles coincide, it may be necessary to reverse, that is, turn over one of them before superposition.

C F

B

D

EXERCISES.

1. Shew that the bisector of the vertical angle of an isosceles triangle (i) bisects the base (ii) is perpendicular to the base.

2. Let O be the perpendicular to it.

middle point of a straight line AB, and let OC be Then if P is any point in OC, prove that PA PB.

3. Assuming that the four sides of a square are equal, and that its angles are all right angles, prove that in the square ABCD, the diagonals AC, BD are equal.

4. ABCD is a square, and L, M, and N are the middle points of AB, BC, and CD: prove that

(i) LM = MN.

(iii) AN=AM.

(ii) AM=DM.

(iv) BN DM.

[Draw a separate figure in each case.]

AB, AC two

5. ABC is an isosceles triangle: from the equal sides equal parts AX, AY are cut off, and BY and CX are joined. Prove that BY= CX.

THEOREM 5. [Euclid I. 5.]

The angles at the base of an isosceles triangle are equal.

brsects Vertical

Δ

B

слід

Let ABC be an isosceles triangle, in which the side AB = the side AC.

It is required to prove that the LABC = the LACB.

Suppose that AD is the line which bisects the BAC, and let it meet BC in D.

1st Proof.

because

Then in the ▲ BAD, CAD,

BA = CA,

AD is common to both triangles,

and the included BAD = the included CAD ;

.. the triangles are equal in all respects; Theor. 4. ABD = the L ACD.

so that the

.. B must fall on C, and consequently DB on DC.

.. the ABD will coincide with the ACD, and is therefore equal to it.

Q.E.D.

COROLLARY 1. If the equal sides AB, AC of an isosceles triangle are produced, the exterior angles EBC, FCB are equal; for they are the supplements of the equal angles at the base.

E

COROLLARY 2. If a triangle is equilateral, it is also equiangular.

DEFINITION. A figure is said to be symmetrical about a line when, on being folded about that line, the parts of the figure on each side of it can be brought into coincidence.

The straight line is called an axis of symmetry.

That this may be possible, it is clear that the two parts of the figure must have the same size and shape, and must be similarly placed with regard to the axis.

Theorem 5 proves that an isosceles triangle is symmetrical about the bisector of its VERTICAL angle.

An equilateral triangle is symmetrical about the bisector of ANY ONE of its angles.

EXERCISES.

1. ABCD is a four-sided figure whose sides are all equal, and the diagonal BD is drawn: shew that

=

(i) the angle ABD the angle ADB ; (ii) the angle CBD the angle CDB; (iii) the angle ABC= the angle ADC.

2. ABC, DBC are two isosceles triangles drawn on the same base BC, but on opposite sides of it. prove (by means of Theorem 5) that the angle ABD = the angle ACD.

3. ABC, DBC are two isosceles triangles drawn on the same base BC and on the same side of it: employ Theorem 5 to prove that the angle ABD the angle ACD.

4. AB, AC are the equal sides of an isosceles triangle ABC; and L, M, N are the middle points of AB, BC, and CA respectively prove

that

THEOREM 6. [Euclid I. 6.]

If two angles of a triangle are equal to one another, then the sides which are opposite to the equal angles are equal to one another.

A

Let ABC be a triangle in which

the ABC = the ACB.

It is required to prove that the side AC = the side AB.

If AC and AB are not equal, suppose that AB is the greater. From BA cut off BD equal to AC.

Proof.

because

.. the

Join DC.

Then in the ▲ DBC, ACB,

DB = AC,

BC is common to both,

and the included DBC = the included ▲ ACB;

DBC the ACB in area,

==

the part equal to the whole; which is absurd.

.. AB is not unequal to AC;

that is, AB AC.

=

Theor. 4.

Q.E.D.

COROLLARY. Hence if a triangle is equiangular it is also

equilateral.

NOTE ON THEOREMS 5 AND 6.

Theorems 5 and 6 may be verified experimentally by cutting out the given A ABC, and, after turning it over, fitting it thus reversed into the vacant space left

in the paper.

B

Suppose A'B'C' to be the original position of the ▲ ABC, and let ACB represent the triangle when reversed.

In Theorem 5, it will be found on applying A to A' that C may be made to fall on B', and B on C'.

In Theorem 6, on applying C to B' and B to C' we find that A will fall on A'.

In either case the given triangle reversed will coincide with its own "trace," so that the side and angle on the left are respectively equal to the side and angle on the right.

NOTE ON A THEOREM AND ITS CONVERSE.

The enunciation of a theorem consists of two clauses. The first clause tells us what we are to assume, and is called the hypothesis; the second tells us what it is required to prove, and is called the conclusion. For example, the enunciation of Theorem 5 assumes that in a certain triangle ABC the side AB=the side AC: this is the hypothesis. this it is required to prove that the angle ABC= the angle ACB: this is the conclusion.

From

If we interchange the hypothesis and conclusion of a theorem, we enunciate a new theorem which is called the converse of the first. For example, in Theorem 5

it is assumed that

AB = AC;

it is required to prove that the angle ABC= the angle ACB.}

Now in Theorem 6

AB AC.

B;}

it is assumed that the angle ABC the angle ACB it is required to prove that Thus we see that Theorem 6 is the converse of Theorem 5; for the hypothesis of each is the conclusion of the other.

In Theorem 6 we employ an indirect method of proof frequently used in geometry. It consists in shewing that the theorem cannot be untrue; since, if it were, we should be led to some impossible conclusion. This form of proof is known as Reductio ad Absurdum, and is most commonly used in demonstrating the converse of some foregoing theorem. It must not however be supposed that if a theorem is true, its con verse is necessarily true. [See p. 25.]