Cor. 1. Hence in a triangle if one angle be right the other two are together to a right angle; and if one angle be to the other two, it is a right angle.

Cor. 2. If two triangles have two angles the remaining angles are also equal.

to two,

Cor. 3. In a right angled isosceles triangle, each angle at the base is half a right angle.

Cor. 4. Each angle of an equilateral triangle is to a third part of two right angles.

Cor. 5 Hence can be derived a method of trisecting a right angle; by assuming any portion on one of the legs from the vertex and constructing on it an equilateral triangle, and bisecting the angle of it cut off from the given right angle.

Cor. 6. All the internal angles of any rectilineal figure together with four right angles are to twice as many right angles as the figure has sides.

For if you assume any point within the given figure and draw lines from each of the angles to it, the figure will be divided into as many triangles as it has sides; and therefore all their angles taken together are to twice as many right angles as the figure has sides, (prop. 23,). but still the angles at the assumed point are to four right-angles (cor. 3. prop. 13.) .. the angles of the figure together with four right angles are &c. &c.

This also can be proved by drawing lines from any angle of the figure to all the others except the two adjacent ones, then the figure will be divided into as many triangles as it has sides 2 and the angles of each of the triangles into which it is divided are to 2. right angles,

&c.

Cor. 7. The external angles of any rectilineal figure are together to four right angles. For each external angle, with the adjacent internal, is to two right angles, (prop. 13.) Therefore, all the external angles, with all the internal, are to twice as many right angles as the figure has sides; but the internal angles, with four right angles, are to twice as many right angles as the figure has sides, take away from both the internal angles, and the external angles remain to four right angles.

PROP. 33. THEOR.

Right lines which join the adjacent extremities of equal and parallel right lines are also themselves equal and parallel,

Draw a diagonal, then two triangles are formed, having a given side, and diagonal in one respectively = to the other given side, and same diagonal in the other, and the contained angles as being alternate, ・. the third sides, or joining lines are and also the angles at them; and since the angles contained by them and diagonal are, the joining lines are parallel.

PROP. 34, THEOR.

The opposite sides of any parallelogram are equal to one another, as are also the opposite angles, and the parallelogram itself is bisected by the diagonal.

=

For in the two triangles formed by the diagonal; the alternate angles are respectively to one another, (prop. 29,) and the side adjacent to the angles, viz. the diag: is common to both triangles, .. the other sides are respectively (prop 26,) viz. the opposite sides; and the triangles themselves are, and the opposite angles which are subtended by the diagonal are, also the other pair of opposite angles must be since either of them is a supplement to either of the former of two right angles. Cor. 1. If one angle of a parallelogram be a right angle, the other three are right angles. For the adjacent angle is right, because with a right angle it is to two right angles, and the opposite are right angles, because they are to these right angles.

Cor. 2. If two parallelograms have one angle in each, the other angles are respectively =. For the angles opposite to these angles are to them, and therefore to each other (ax. 2.) but the angles adjacent to them are also to each other, as being together with these equal angles to two right angles. (prop. 29.)

PROP. 35, THEOR.

Parallelograms on the same base and between the same parallels are equal.

Connect the sides opposite the base, (if necessary,) then there are two triangles formed (viz. the excesses of the quadrilateral figure above each parallelogram,) having two sides of one respectively to two sides of the other, and the contained angles since they are the differences of angles (two of which are formed by producing the common base) .. the triangles are; if one of those triangles be taken from the whole figure it will leave one par, and the other being taken away leaves the other par. which parallelograms must.. be.

PROP. 36, THEOR.

Parallelograms upon equal bases and between the same parallels are equal.

For, if the extremities of the base of one of them be connected, by right lines, with the adjacent extremities of the side opposite the base of the other, those connecting lines with the connected lines form a parallelogram to either of them ... they are to one another.

PROP. 37, THEOR.

Triangles on the same base and between the same parallels are equal.

Draw through one extremity of the base, (to meet the line parallel to the base,) lines respectively par. to the sides of the triangles terminating in the other extremity of the base; thus parallelograms are formed (35. 1.) of which the given triangles are halves (34. 1.) and are .. equal.

=

PROP. 38, THEOR.

Triangles on equal bases, and between the same parallels are equal,

Draw through either extremity of each base (to meet the line par. to the bases) lines par. to the sides of the triangles terminating in the other extremity of each base; thus there are two parallelograms formed of which the given triangles as halves they are., equal.

PROP. 39, THEOR.

Equal triangles on the same base and on the same side of it are between the same parallels.

For if they are not, draw through the vertex of one of them a line par. to the base, it cuts a side of the other or it produced, connect the point of section with the opposite angle, the triangle so formed shall be to a triangle (37. 1.) which is to a triangle that must be either greater or less than this, which is absurd; .. they must be between the same par.

PROP. 40, THEOR.

Equal triangles on equal bases and on the same side are between the same parallets.

This proposition may be demonstrated precisely as the former, (from the 38th.)

PROP. 41, THEOR.

If a parellogram and a triangle have the same base and are between the same parallels the parallelogram is double of the triangle.

For by drawing a diagonal you divide the parallelogram, into two triangles, either of which is to the given triangle,.. the parellelogram is double of the triangle.

Schol. Hence it is evident that a parallelogram is double of a triangle if they have bases and are between the same parallels.

Cor. If a parallelogram and a triangle be between the same parallels and the base of the triangle be double that of the parallelogram, the parallelogram is to the triangle.

For if the base of the triangle be bisected and the point of bisection be connected with the vertex, it is di vided into triangles, either of which is half the parallelogram, .. the parallelogram is to the whole triangle.

PROP. 42, PROB.

To construct a parallelogram equal to a given triangle, and having an angle equal to a given one.

Through the vertex of the triangle draw a right line par. to the base, bisect the base and at the point of section; with either segment draw a line, (to meet the line parallel to the base.) making with the segment an angle = to the given one, and through the other extremity of this seg. draw a line par. to the one making the required angle, this parallelogram thus formed is to the given triangle: for it is double of half the triangle, (41. 1.) and it has an angle to the given one.

PROP. 43, THEOR.

In a parallelogram the complements of the parallelograms about the diagonals are equal.

For the whole parallelogram is divided (by the diagonal) into equal triangles, .. if from each of the large triangles you take away the small ones which respectively belong to them, the complements will remain =.

By the complements are meant those parallelograms which with the ones about the diagonal make up the whole parallelogram.

Cor. The parallelograms about the diagonal of a square are squares.