Triangle

TRIANGLE. A triangle, or more precisely a plane triangle, is the geometrical figure composed of three points called the vertices (not lying in one straight line), and the three straight lines joining these, called the sides. Since no part of a plane can be inclosed by fewer than three straight lines, the triangle ranks as the simplest figure of its class, and plays a correspondingly important part in both practical and theoretical geometry.

Congruence Theorems.

It is important to recognize the conditions under which two triangles may be known to be con gruent; that is, alike in all respects except position in space, so that the one triangle is merely the other moved into a new posi tion without change of form. The most important theorems of this character have been known from very ancient times, and were handed down in the first book of Euclid's Elements.

Two triangles are congruent if two sides of the one are respec tively equal in length to two sides of the other, and if these two sides are in each case inclined at the same angle.

Thus in figure I, to be sure that the triangles ABC and abc are exactly alike, it is enough to know that AB=ab, AC=ac, and the angle at A =the angle at a. It would not be enough to know that these same two pairs of sides were equal together with the angles at B and b, since in this case the triangle ABC might equally well have had the form of ABC' in the figure, and therefore not have agreed with abc in form.

Two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other.

Two triangles are congruent if a side of one is equal to a side of the other, and if the angles at the ends of these equal sides are the same in both triangles.

Thus the triangles ABC and abc will be identical in form if BC=bc, the angle at B=the angle at b, and the angle at C=the angle at c.

This theorem supplies the basis for the method of triangula tion. If B and C are given stations whose position is known, the measurement of the angles ABC and ACB will determine, together with a knowledge of BC, the form of the triangle ABC, and so the position of A.

Two triangles are congruent if two angles of the one are equal to two angles of the other respectively, and if the sides opposite one of the equal pairs of angles are equal.

This theorem is readily reduced to the one before by means of the property stated at the beginning of the next section. The same consideration prepares us to accept the statement that two triangles may have the angles of one equal to the angles of the other without being congruent. In such a case the triangles are similar in shape, but may be of different sizes.

Relations Between the Parts of a Triangle.

The foregoing remarks emphasize the fact that the three sides and the three angles are not six independent magnitudes to which any values whatsoever may be given. For instance it is impossible to draw a

triangle in which all the sides are equal and the angles unequal.

The simplest relation connecting the parts of a triangle in general is that the sum of the three angles is two right angles. This follows from the fact that, when one side is continued out side the triangle, the angle so formed on the outside is equal to the sum of the angles inside the triangle at the other two vertices. The latter relation is exhibited in figure I for the triangle ABC, the line CD, drawn parallel to BA, dividing the exterior angle into two parts, of which DCA' is equal to A, and DDC' to B. The relation between the three angles enables one to find the value of the third when two of the angles are known. Thus, if two of the angles are 6o° and 7o°, the third is 18o°-6o°-7o° = 5o°.

It is convenient to consider here certain special forms of triangle. A triangle with two equal sides is called isosceles. Such a triangle has the angles opposite to these sides also equal. This is easily proved by joining the middle point of the unequal side to the opposite vertex and deducing the congruence of the two triangles so formed. The proof given by Euclid was much more elaborate, be cause Euclid did not permit the use of the middle point at a stage when the problem of finding it had not been taken up ; and the theorem was nick-named the pons asinorum, and regarded at one time as a notable obstacle to the beginner in geometry.

A right triangle is one which has one of its angles a right angle. In figure 2 the triangle ABC has a right angle at C. Evidently CB=AB x s, and AC=AB x k, where s and k are proper frac tions depending on the angle A, and not on the size of the triangle.

The ratios s and k are called the sine and cosine respectively of the angle A, and their theory constitutes the science of trigonom etry (q.v.). If CD is perpendicular to AB, by the similarity of the three triangles, BD =CBXs= and AD = ACxk= AB Since AD -I- BD=AB, it follows that On multiplying this equation by it is seen that Here the meaning of may be understood to be the arithmetical square of the number of units in BC. The side AB opposite the right angle is called the hypotenuse of the right tri angle, and the result just obtained is that the square of the hypot enuse is equal to the sum of the squares of the other two sides, a theorem ascribed to Pythagoras (q.v.), and of the utmost im portance in geometry.

Allusion may be made to two triangle inequalities. (I) The sum of any two sides is greater than the third side; (2) the. greater of two unequal sides has the greater angle opposite to it.

As supplementing the preceding statement (2) we have the sine formula. In fig. 3, if CR is perpendicular to AB, the sines of A and B are respectively RC — and — RC • These are proportional to