Similar figures are figures which have the same shape.
Equal figures are figures which have the same size.

Congruent figures are figures which have both the same shape and the same size. Equal figures which are not congruent are called equivalent.

The congruence of two figures may be tested by the method of superposition. If the figures coincide when one is placed upon the other, they are congruent.

In congruent or similar figures the corresponding parts are called homologous.

Homologous parts of congruent figures are equal.

Parallel lines are lines lying in the same plane which will never meet even if produced indefinitely.

A polygon is a plane closed figure bounded by straight lines. The sum of the bounding lines is called the perimeter.

A line joining two vertices of a polygon which are not consecutive is called a diagonal.

A polygon of three sides is called a triangle; one of four sides, a quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of eight sides, an octagon; one of ten sides, a decagon; one of twelve sides, a dodecagon, and so on.

An isosceles triangle has two sides equal; an equilateral triangle has all three sides equal.

A parallelogram is a quadrilateral which has its opposite side parallel.

A rectangle is a parallelogram whose angles are right angles. A square is a rectangle whose sides are equal.

Two polygons are similar when their homologous angles are equal and their homologous sides are proportional.

Homologous angles of similar polygons are equal.

Homologous sides of similar polygons are proportional.

A circle is a closed curve lying in a plane such that all points of the curve are equally distant from a fixed point in the plane called the centre.

When a variable, which changes according to some fixed law, can be made to have values such that the difference between the variable and a certain constant becomes and remains less than any assigned positive quantity, however small, the variable is said to approach the constant as a limit, and the constant is called the limit of the variable.

Ax. 1. Quantities equal to the same quantity, or to equal quantities, are equal to each other.

Ax. 2. A quantity can be substituted for its equal in an equality or in an inequality.

Ax. 3. If three quantities are so related that the first is greater than the second, while the second is greater than the third, then the first is greater than the third.

Ax. 4. If equals are added to equals, the sums are equal.

Ax. 5. If equals are added to unequals, the sums are unequal in the same order.

Ax. 6. If unequals are added to unequals in the same order, the sums are unequal in the same order.

Ax. 7. If equals are subtracted from equals the remainders are equal.

Ax. 8. If equals are subtracted from unequals, the remainders are unequal in the same order.

Ax. 9. If unequals are subtracted from equals, the remainders are unequal in reverse order.

Ax. 10. If equals are multiplied by equals, the products are equal. Ax. 11. If unequals are multiplied by equals, the products are unequal in the same order.

Ax. 12. If equals are divided by equals, the quotients are equal. Ax. 13. If unequals are divided by equals, the quotients are unequal in the same order.

Ax. 14.

Ax. 15.
Ax. 16.
Post. 1.

only one.
Post. 2.

Post. 3.

Post. 4.

Like powers or like roots of equals are equal.
The whole is greater than any one of its parts.

The whole is equal to the sum of all its parts.

Between two points one straight line can be drawn, and

A straight line can be produced to any length.

A straight line is the shortest distance between two points. Two straight lines can intersect in only one point. Post. 5. A circle can be described about any given point as a centre with a radius of any given length.

Post. 6. A figure can be moved from one position to another without change of size or shape.

Postulate of Parallels. Two intersecting lines cannot both be parallel to the same line.

Prop. 1. At a given point in a given straight line one perpendicular to the line can be erected, and only one.

Prop. 2. All right angles are equal.

Prop. 2, Cor. III. The supplements of equal angles are equal. Prop. 3. If one straight line meets another straight line not at its extremity, the sum of the two adjacent angles is equal to two right angles. Prop. 3, Cor. III. The sum of all the successive angles formed about a point is equal to four right angles.

Prop. 4. If the sum of two adjacent angles is equal to two right angles, their exterior sides lie in the same straight line.

Prop. 5. If two straight lines intersect each other, the vertical angles are equal.

Prop. 7. The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.

Prop. 8. Two triangles are congruent when two sides and the included angle of one are equal respectively to two sides and the included angle of the other.

Prop. 8, Cor. Two right triangles are congruent when the two legs of one are equal respectively to the two legs of the other.

Prop. 9. Two triangles are congruent when a side and the two adjacent angles of one are equal respectively to a side and the two adjacent angles of the other.

Prop. 10. From a given point without a given straight line one perpendicular to the line can be drawn, and only one.

Prop. 11. Two right triangles are congruent when the hypotenuse and an adjacent angle of one are equal respectively to the hypotenuse and an adjacent angle of the other.

Prop. 12. In an isosceles triangle the angles opposite the equal. sides are equal.

Prop. 12, Cor. An equilateral triangle is also equiangular.

Prop. 13. The bisector of the vertical angle of an isosceles triangle bisects the base and is perpendicular to the base.

Prop. 13, Cor. In an isosceles triangle or an equilateral triangle the bisector of the vertical angle, the altitude, the median, and the perpendicular bisector of the base are one and the same straight line.

Prop. 14. If two angles of a triangle are equal, the sides opposite these angles are equal, and the triangle is isosceles.

Prop. 14, Cor. An equiangular triangle is also equilateral.

Prop. 15. Two triangles are congruent when the three sides of one are equal respectively to the three sides of the other.

Prop. 16. Two right triangles are congruent when the hypotenuse and a leg of one are equal respectively to the hypotenuse and a leg of the other.

Prop. 17. If two angles of a triangle are unequal, the side opposite the greater angle is longer than the side opposite the less.

Prop. 18. If two sides of a triangle are unequal, the angle opposite the longer sides is greater than the angle opposite the shorter.

Prop. 19. If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, the third side of the first is greater than the third side of the second.

Prop. 20. If two triangles have two sides of one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second, the angle opposite the third side of the first is greater than the angle opposite the third side of the second.

Prop. 22. If two oblique lines drawn from a point to a line meet the line at equal distances from the foot of the perpendicular drawn from the point to the line, they are equal.

Prop. 22, Cor. I. Every point in the perpendicular bisector of a line is equidistant from the extremities of the line.