Mutual Equality of Triangles.

When the three sides of one triangle are respectively equal to the three sides of another, the angles opposite to equal sides are also equal, and the triangles are equal.

When two triangles are mutually equiangular, and have two corresponding sides equal, the other corresponding sides (or the sides opposite to equal angles in each triangle) will be equal.

When two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another, then the third sides are equal, and the angles opposite to equal sides are also equal,

Properties of Triangles.

In every triangle the greater of any two sides subtends the greater angle; and, conversely, the greater of any two angles is subtended by the greater side.

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

If one side of any triangle be produced, the external angle is equal to both the internal and opposite angles. The sum of the three angles of every triangle is equal to two right angles.

If two angles of one triangle be together equal to two angles of another, the third angle of the former triangle is equal to the third angle of the latter.

If one angle of any triangle be a right angle, the sum of the other two angles is equal to a right angle.

If one angle of any triangle be either right or obtuse, each of the other two angles is acute.

Triangles on the same base, or on equal bases, and between the same parallels, are equal.

If two sides of a triangle be equal, the opposite angles are equal; and if two angles of a triangle be equal, the opposite sides are equal,

Properties of Parallelograms.

Straight lines which join the corresponding extremities of two equal and parallel straight lines are also equal and parallel; that is, those four lines make a parallelogram.

The opposite sides of a parallelogram are equal, and also the opposite angles.

The diagonal of a parallelogram divides it into two equal triangles.

If the opposite sides of a quadrilateral figure be equal, they are also parallel; and if the opposite angles of a quadrilateral figure be equal, the opposite sides are parallel; that is, in both cases the figure is a parallelogram.

The diagonals of any parallelogram bisect each other.

Parallelograms on the same base, or on equal bases, and between the same parallels, are equal.

If a parallelogram and a triangle stand on the same base and between the same parallels, the parallelogram is double of the triangle.

The complements of the two parallelograms which are about the diagonal of any parallelogram are equal

to each other.

General Properties of Rectilinear Figures.

The sum of all the internal angles of any rectilinear figure is equal to twice as many right angles, except four, as the figure has sides.

If all the sides of any rectilinear figure be produced outward, the sum of all the external angles is equal to four right angles.

The sum of the four angles of every quadrilateral figure is equal to four right angles.

APPENDIX.

The demonstrations of the properties of parallel lines are seldom understood by learners. Indeed the subject of parallel lines is allowed to be one of the most difficult in the elements of geometry, and has exercised the ingenuity of many modern geometers, who have not been able to remove all the difficulties without impairing the rigour of geometrical demon'stration. Some have attempted to give more simple demonstrations than those of Euclid, by means of new axioms, and others by means of a number of auxiliary propositions.

The following method of demonstrating the properties of parallel lines is plain and intelligible to youth, and appears to be legitimate. The definition of parallel lines which is here adopted has been proposed by some skilful mathematicians, and seems to be unexceptionable.

Definition. Parallel lines are such as lie in the same plane, and have no inclination to one another; that is, if produced to any length both ways they neither approach to nor recede from one another.

If EF be perp. to one AB, it will be perp. to the other CD; if it cross one obliquely, it will also cross the other with the same obliquity.

PROPOSITION I. THEOREM.

A straight line intersecting two parallel straight lines makes the alternate angles equal, and the two interior angles on the same side of it together equal to two right angles.—See figure on next page.

Let the straight line EF intersect the two parallel straight lines AB, CD; the alternate angles AGH, GHD are equal, and the angles BGH, GHC are equal; and the two interior angles BA, GHD, on the same side of EF, are together equal to two right angles.

Since AB, CD are parallel, EF has the same inclination to each of them, or the angles EGB, EHD are equal (2 Cor. Def.); and because the two lines AB, EF intersect each other, the vertical angles AGH, EGB are equal (Prop. 15); consequently the angle AGH is equal to GHD. In like manner the angle BGH is equal to GHC.

Again, the angles BGE, BGH are together equal to two right angles (Prop. 13), and the angle BGE is equal to GHD; therefore the angles BGH, GHI) are together equal to two right angles. Therefore a straight line &c. Q. E. D.

If a straight line intersect two other straight lines, and make the alternate angles equal to each other, or make the two internal angles, on the same side of it,

together equal to two right angles; then those two lines are parallel to each other.

Let a straight line EF intersect two other straight lines AB, CD, and make the alternate angles equal, AGH=GHD, and BGH GHC; or make the internal angles BGH, GHD, on the same side of it, together equal to two right angles; AB is parallel to CD.

For since the angle BGE is equal to AGH (Prop. 15), it is also equal to the angle GHD; therefore AB is parallel to CD (2 Cor. Def.).

Again, because the angles BGH, GHD are together equal to two right angles, and the angles BGH, BGE are together equal

to two right angles (Prop. 13); the angle BGE is equal to GHD; therefore AB is parallel to CD. Therefore, if a straight line &c. Q. E. D.

Otherwise. For if they were not parallel the alternate angles would not be equal, nor the two internal angles on the same side together equal to two right angles.

Cor. Two straight lines AB, CD, which are perpendicular to the same straight line EF, are parallel to each other.-See the second figure.

1

For the angles AEF, EFD are equal, because they are right angles. But they are alternate angles, therefore AB is parallel to CD.

PROPOSITION III. THEOREM.

Straight lines which are parallel to the same straight line are parallel to one another.

Let two straight lines AB, CD be parallel to the same straight line EF; they are parallel to each other.