Schol. It is evident that the remaining angles opposite the = sides are also and also the triangles themselves.

PROP. 9, PROB.

To bisect a given rectilineal angle.

On the sides containing the angle assume portions towards the vertex, connect their extremities and on the side of the connecting line remote from the given angle, describe an equilateral triangle, the right line joining its vertex with that of the given angle bisects said angle.

For thus there are two triangles formed, having a com mon side, (viz. line connecting vertices,) and assumed portion in one, respectively to said common side and assumed portion in the other, and also the bases or third sides = (viz. sides of equilat. trian.).. the angles contained by the common side and assumed portions are = (per prop 8,) which are the parts of given angle. .. &c. Cor. By this prop. an angle can be divided in 4, 8, 16 &c. parts, by bisecting each part.

PROP. 10. PROB.

To bisect a given finite right line.

Describe on the given right line an equilateral triangle, bisect the angle opposite the given line, the line bisecting this angle, (if produced to it,) will bisect the given right line.

For thus there are two triangles formed, having the bisecting line and side of equilateral triangle in one respectively to said bisecting line, and an other side of equilat. trian. in the other, and the contained angles = (by construction.) .. the bases are = (by 4th,) .. the given right line is bisected.

PROP. 11, PROB.

From a given point in a given right line to draw a perpendicular to the given line.

por

From the given point (on each side of it,) assume tions, upon their sum describe an equilateral triangle and

connect its vertex with the given point; the connecting line is at right angles to the given line.

For thus there are formed two triangles having the three sides of one respectively to those of the other, ..., &c. Schol. In the same manner a perpendicular can be erected at one extremity of a given line, by first producing the line.

PROP. 12, PROB.

To draw a perpendicular to a given indefinite right line from a given point without it.

Assume any point on the other side of the line and from the given point as centre with the assumed point as radius describe a circle, cutting the given line in two points, bisect the intercept and connect the given point with the extremities of it, and also with the point of bisection, this line drawn to the point of bisection is perpendicular to the given line.

This can be proved as the former (per 8th)

PROP. 13, THEOR.

When a right line standing upon another makes angles with it, they are either two right angles or are together equal to two right angles,

If the right line standing on the other be perpendicular to it, the angles are both right; if not, from the vertex draw a perpendicular (per 11.) it is evident that the angles thus formed are to the given angles: therefore, &c.

Cor. 1. If several right lines stand on the same right line at the same point, they make angles, which taken together are equal to two right angles.

Cor. 2. Two right lines intersecting one another make angles, which taken together are to four right angles.

Cor. 3. If several right lines intersect one another at the same point all the angles taken together are to four right angles.

PROP. 14, THEOR.

If two right lines meeting another right line at the same point and at opposite sides, make angles with it, which are together equal to two right angles, those two right lines form one continued right line.

For let it be supposed that one of them be produced beyond the vertex of the angles so as that the produced part shall not coincide with the other: then two angles will be formed to two right angles (by 13,) and those are also either greater or less than the two given angles; which is absurd.

PROP. 15, THEOR.

If two right lines intersect one another the vertical angles are equal.

For they have a common supplement to two right angles.

PROP. 16, THEOR.

If one side of a triangle be produced the external angle is greater than either of the internal opposite angles.

For connect with the opposite angle the middle point, of the side conterminous with the produced part and produce this connecting line till the produced part is to it, and draw a right line from the extremity of the produced part to the external angle, this line shall form with the bisected side of the triangle an angle (subtended by the produced part of bise ting line) to the angle of the given triangle subtended by the other part of bisecting line. (For the triangles to which these angles thus subtended belong have an angle in each as being vertically opposite and the sides containing them by con struction) But the external angle is greater than the former angle and.. greater than the latter. This exter

nal angle is proved greater than the other internal remote angle by producing the opposite side beyond the vertex of this external angle, and proving as before the new external angle, thus formed greater than that internal angle: but this external angle is to the given one, (by the 15th.)

Cor. 1. If from any point two right lines be drawn to the same right line, one of them perpendicular to it and the other not, the perpendicular falls at the side of the acute angle

For if it fell at the side of the obtuse angle then an acute angle would be greater than a right, which is absurd

Cor. 2. Two perpendiculars cannot be drawn from the same point to the same right line.

For then an external angle would be to an internal and opposite which is absurd.

PROP. 17, THEOR,

Any two angles of a triangle are together less than two right angles.

Produce any side, then the external angle is greater than either of the internal remote angles; but the external with the adjacent internal are to two right angles. this internal with either of the other internal angles are less than two right angles. Thus any two angles of a triangle can be proved less than two right angles.

Cor. 1. If any angle of a triangle be obtuse or right, the other two are acute; and if two angles of a triangle be equal to one another they are acute.

PROP. 18, THEOR.

In any triangle if one side be greater than another the angle opposite the greater side is greater than the angle opposite the less.

From the greater side cut off a part to the less and conterminous with it, and connect the extremity of the

part cut off with the opposite angle. This connecting line will cut off from the angle opposite the greater side a part to an angle which is greater than the angle opposite the lesser side (as being external to it by the 16th.) the entire angle opposite the greater side is much greater than the angle opposite the less.

PROP. 19, THEOR.

If in any triangle one angle be greater than an other, the side which is opposite the greater angle is greater than the side which is opposite the less.

For, if it be not greater it must be either to it, or less than it, is not to it; for then the angles subtended by them would be, nor is it less than it; for then the angle opposite to it, would be less than the other, .. it must be greater.

Cor. If from the same point two right lines, be drawn to the same right line, of which one is perpendicular to it, and the other not, the perpendicular is the least.

PROP. 20, THEOR.

Any two sides of a triangle are together greater than the third.

Bisect the angle contained by the two sides, the bisecting line shall cut the third side into segments which are severally less than their conterminous sides since they are opposite to lesser angles, (for the angle subtended by either side is greater than the angle subtended by the conterminous segment, since it is greater than the other half of bisected angle, as being external to it.)

Hence it is evident that the difference between two sides of a triangle is less than the third side.

PROP. 21, THEOR.

Two right lines drawn to any point within a triangle from the extremities of any side, are together less than the sum of the other two sides of the triangle, but contain a greater angle.

Produce either of the lines drawn within the triangle to meet one of the sides, then the whole produced line is

B

་