less than the uncut side and conterminous segment of cut side, to each add the remaining seg. then this produced line and added seg. are less than the sides of original triangle, it also can be proved in the same way that the two drawn lines are less than the produced line and added seg... a fortiori &c. Secondly, the angle contained by those drawn lines is external to an angle that is external to the angle contained by the two sides of 'original triangle and ... much greater than it.

PROP. 22, PROB.

Given three right lines of which any two together are greater than the third to construct a triangle, whose sides shall be respectively equal to the three given lines.

From any point draw a right line to one of them and from the extremities of this, draw right lines réspectively to the other two, and with these extremities as centres, and the lines drawn from them as intervals describe two circles, and from either of their intersections draw lines to the centres (or the extremities of the first drawn line) thus the triangle is formed.

For those lines drawn from intersection are respectively to the two last assumed lines as being radii of the same circles, .. the three sides of the triangle are respectively to three given lines.

Cor. In this manner we may construct a triangle = to a given one.

NOTE.

It is evident that the circles must intersect, from the limitation of the problem.

PROP. 23, PROB.

At a given point in a given right line to make an angle equal to a given angle.

In the legs of the given angle take any points and connect them, then construct on the given line a triangle to the one thus formed, having the sides about the angle at the given point respectively to those about the given angle.

Those sides will contain an angle to the given angle. (by the 8th.)

PROP. 24, PROB.

If two triangles have two sides of the one respectively equal to two sides of the other, and if one of the angles contained by the equal sides be greater than the other, the side opposite the greater angle is greater than the side opposite to the lesser angle.

From the vertex of the greater angle, draw a line making with the side which is not the greater* an angle to the lesser; make this drawn line to the side which is not the lesser, connect its extremity with the extremity of the side which is not the greater, this connecting line will be to the side subtending the lesser angle (by 4th.) But it is less than the line subtending the greater angle, for if you join the extremities of it and the line subt nding the greater angle, it shall be opposite a smaller angle, (viz. to part of an angle at the base of an isosceles triangle, whilst the side subtending the greater angle, is op posite to an angle greater than the other angle of the isosceles triangle at the same base.)

The reason why the angle is made with the side which is not the greater, is fully explained in the notes of Dr. Elrington's Euclid, page, 150.

PROP. 25. THEOR.

If two triangles have two sides of the one respectively equal to two sides of the other, and if the third side of the one be greater than the third side of the other, the angle opposite the greater side is greater than the angle which is opposite the lesser.

For if it be not greater, it is either to it, or less than it; it is not, for then the sides opposite to them would be, nor is it less, for then the side which is given greater, would be less than the one given less. Therefore, since it is neither nor less, it must be greater.

PROP. 26, THEOR.

two triangles have two angles of the one respeotively equal to two angles of the other, and a side of the one equal to a side of the other, whether it be adjacent or opposite to these equal angles, the remaining sides and angles are respectively equal.

angles in

First, let the side be adjacent to the each, the sides opposite to either of the angles shall be, for if not cut off from the greater, a part = to the less and conterminous with the given side; and draw a right line from the extremity of that part, to the vertex of the opposite angle; this drawn line will cut off from the angle to which it is drawn, an angle to that angle of the other triangle, which is to the entire of this cut angle, which is absurd. 2nd, Let the sides be opposite to the angles; it may in like manner be proved, that the sides between the angles are, for if they were not. an external angle of a triangle would be = to an internal remote.

Schol. It is evident that the triangles themselves are equal.

Cor, 1. In an isosceles triangle, the right line drawn from the vertex perpendicular to the base, bisects the base and vertical angle.

For, the triangles thus formed have two angles, and one side in each respectively. the other sides and angles are respectively, viz. the segs. of base; and of vertical angle.

Cor. 2, It is evident that the right line which bisects the vertical angle of a triangle, bisects the base and is perpendicular to it, and that the right line drawn from the vertical angle bisecting the base is perpendicular to it, and bisects the vertical angle.

PROP 27, THEOR.

If a line intersect two right lines and makes the alternate angles equal to each other, these two right lines are parallel.

For if not, let them meet on either side, then the ex, ternal angle of the triangle is greater than the internal

remote, but it is also to it which is absurd. .. the lines will not meet if produced this way, and in like manner it can be proved that they will not meet if produced the other way, .. they are parallel.

PROP. 28, THEOR.

If a right line intersect two right lines and makes the external angle equal to the internal and opposite on the same side of the line, or makes the internal angles on the same side together, equal to two right angles, the two right lines are parallel to one another.

For, in the first case, the alternate angles are, as one of them is to the external angle (by the 15th,) ... the lines are parallel. And in the second case, the alternate angles are, since one of them with the adjacent internal is to two right angles, and this adjacent internal with the other internal, also to two right angles... &c.

PROP. 29, THEOR.

If a right line intersect two parallel right lines, it makes the alternate angles equal; and the external angle equal to the internal, or opposite on the same side, and also the two internal angles at the same side together equal to two right angles.

For, if the alternate angles be not, let one of them if possible be greater than the other, then if the lines be produced towards the side of the lesser angle, they will meet, since it, with the internal on the same side would then be less than two right angles, (by ax 12,) but since the lines are parallel they cannot meet, .. the alternate angles are not unequal. And since the alternate angles are, the external which is to one of them (by the 15th,) is = to the internal and opposite on the same side. Also, since the alternate angles are, the two internal angles are together to two right angles, for one of the alternate angles with the adjacent internal are to two right angles, (by the 13th.).. &c.

PROP. 30, THEOR.

If two right lines be parallel to the same right line, they are parallel to one another.

Let a right line be drawn cutting the three given lines, this drawn line makes with each pair of parallel lines an external angle = to an internal or remote on the same side of the line, and.. does so with the pair to be proved parallel,

PROP. 31, PROB.

Through a given point to draw a right line parallel to a given right line.

In the given line take any point, connect the given point with it, and through the given point draw a line making an angle with the connecting line to an angle made by connecting line with given line, but on the contrary side of it. This second drawn line is parallel to the given line, since the alternate angles are equal.

PROP. 32, THEOR.

If any side of a triangle be produced, the external angle is equal to the sum of the two internal and opposite angles, and the three internal angles of any triangle taken together, are equal to two right angles.

Draw through the external angle, a line parallel to the side, not conterminous with the produced part, (i. e. to the side opposite the adjacent i ternal angle ;) it divides the external angle into two parts respectively = to the internal and remote angles, (by the 29th,) therefore, &c. And since the external angle is to the two internal and remote angles and the external with the adjacent internal to two right angles the adjacent internal, with the two internal remote angles are together to two right angles.