AC to DF, and the angle A to the angle D: then are the triangles equal in all respects.

For, let ABC be applied to DEF, in such a manner that the angle A shall coincide with the angle D, the side AB taking the direction DE, and the side AC the

direction DF. Then, because AB is equal to DE, the vertex B will coincide with the vertex E; and because AC is equal to DF, the vertex C will coincide with the vertex F; consequently, the side BC will coincide with the side EF (A. 11). The two triangles, therefore, coincide throughout, and are consequently equal in all respects (I., D. 15); which was to be proved.

PROPOSITION VI. THEOREM.

If two triangles have two angles and the inciuded side of the one equal to two angles and the included side of the other, each to each, the triangles are equal in all respects..

In the triangles ABC and DEF, let the angle B be equal to the angle E, the

angle C to the angle F,

and the side BC to the

side EF: then are the triangles equal in all re- B

spects.

CE

For, let ABC be applied to DEF in such a manner that the angle B shall coincide with the angle E, the side BC taking the direction EF, and the side BA the direc

tion ED. Then, because BC is equal to EF, the vertex C will coincide with the vertex F; and because the angle C is equal to the angle F, the side CA will take the direction FD. Now, the vertex A being at the same time on the lines ED and FD, it must be at their intersection D (P. III., C.): hence, the triangles coincide throughout, and are therefore equal in all respects (I., D. 15); which was to be proved.

PROPOSITION VII. THEOREM.

The sum of any two sides of a triangle is greater than the third side.

For, the distance from A to C, measured

on any broken line AB, BC, is greater than

the distance measured on the straight line AC (A. 12): hence, the sum of AB and BC is greater than AC; which was to be proved.

Cor. If from both members of the inequality,

ACAB+ BC,

we take away either of the sides AB, BC, as BC, for example, there remains (A. 5),

ACBC AB;
<

that is, the difference between any two sides of a triangle is less than the third side.

Scholium. In order that any three given lines may rep

resent the sides of a triangle, the sum of any two must be greater than the third, and the difference of any two must be less than the third.

PROPOSITION VIII. THEOREM.

if from any point within a triangle two straight lines are drawn to the extremities of any side, their, sum is less than that of the two remaining sides of the triangle.

Let O be any point within the triangle BAC, and let the lines OB, OC, be drawn to the extremities of any side, as BC: then the sum of BO BO and OC is less than the sum of the sides BA and AC.

B4

Prolong one of the lines, as BO, till it meets the side AC in D; then, from Prop. VII., we have,

OCOD + DC;

adding BO to both members of this inequality, recollecting that the sum of BO and OD is equal to BD, we have (A. 4),

BO + OC < BD + DC.

From the triangle BAD, we have (P. VII.),

BDBA + AD;

adding DC to both members of this inequality, recollecting that the sum of AD and DC is equal to AC, we have,

BD + DC < BA + AC.

But it was shown that BO+ OC is less than BD + DC; still more, then, is BO+OC less than BA + AC; which was to be proved.

If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angles unequal, the third sides are unequal; and the greater side belongs to the triangle which has the greater included angle.

In the triangles BAC and DEF, let AB be equal to DE, AC to DF, and the angle A greater than the angle D: then is BC greater than EF.

Let the line AG be drawn, making the angle CAG equal to the angle D (Post. 7); make AG equal to DE, and draw GC. Then the triangles AGC and DEF have two sides and the included angle of the one equal to two sides and the included angle of the other, each to each; consequently, GC is equal to EF (P. V.).

Now, the point G may be without the triangle ABC, it may be on the side BC, or it may be within the triangle ABC. Each case will be considered separately.

whence, by addition, recollecting that the sum of Bl and IC is equal to BC, and the sum of GI and IA, to GA, we have,

AG + BC > AB + GC.

=

Or, since AG = AB, and GC EF, we have,

AB+ BC > AB + EF.

Taking away the common part AB, there remains (A. 5),

3o. When G is within the triangle ABC.

From Proposition VIII., we have,

BA+BC > GA + GC;

Hence, in each case, BC is greater than EF; which was to be proved.

Conversely: If in two triangles ABC and DEF, the side AB is equal to the side DE, the side AC to DF, and BC greater than EF, then is the angle BAC greater than the angle EDF.

For, if not, BAC must either be equal to, or less than, EDF. In the former case, BC would be equal to EF (P. V.), and in the latter case, BC would be less than EF; either of which would contradict the hypothesis: hence, BAC must be greater than EDF.