In passing counter-clockwise about the perimeter of a polygon the angles on the left are called interior angles of the polygon, or for brevity simply the angles of the polygon.

Such are the angles CBA, DCB, EDC, in the figure on p. 18.

.....

If the sides of a polygon are produced in the same order, the angles between the sides produced and the following sides are called the exterior angles of the polygon.

Such are the angles XBC, YCD, in the figure on p. 18.

A line joining the vertices of any angles of a polygon, which have not a common arm, is called a diagonal.

Such a line would be the one joining A and C in the figure on p. 18. How many diagonals can be drawn in that polygon of five sides? Name them.

The sides, angles, and diagonals of a polygon are often called its parts. A polygon which has all of its sides equal is called equilateral.

A polygon which has all of its angles equal is called equiangular.

Two polygons are said to be mutually equilateral, or one is said to be equilateral to the other, when the sides of the one are respectively equal to the sides of the other.

Two polygons are said to be mutually equiangular, or one is said to be equiangular to the other, when the angles of the one are respectively equal to the angles of the other.

A polygon of three sides is called a triangle; one of four sides, a quadrilateral.

Other special kinds of polygons are mentioned after th. 20.

Any side of a polygon may be called its base, the side on which the figure appears to stand being usually so called.

In the case of a triangle, the vertex of the angle opposite the base is called the vertex of the triangle, the angle itself being called the vertical angle of the triangle, and the other two angles being called the base angles.

Two figures which may be made to coincide in all their parts by being placed one upon the other are said to be congruent. For example, two line-segments may be congruent, or two angles, or two triangles, etc.

The operation of placing one figure upon the other so that the two will coincide is called superposition, and the figures are sometimes called superposable, a synonym of congruent.

Superposition is an imaginary operation. It is assumed as a postulate that figures may be moved about in space with no other change than that of position. The actual movement is, however, left for the imagination.

It will hereafter be explained and defined that polygons of the same shape are called similar, the symbol of similarity being, and that those of the same area are called equal or equivalent, the symbol being =. Congruent figures are both similar and equal, and hence the symbol for congruence is, a symbol used in modified form by the great mathematician Leibnitz. The symbol is derived from the letter S, the initial of the Latin similis, similar.

Many writers use equal for congruent, and equivalent for equal, as above defined. But because of the various meanings of the word equal, and its general use as a synonym for equivalent, the more exact word congruent with its suggestive symbol is coming to be employed.

NOTE. In the exercises hereafter given, the demonstrations of theorems are to be given in full; when a question is asked, a proof of the answer is to be given; when a theorem is suggested, it is to be completely stated and then proved. The exercises given on p. 21 can be proved from th. 1 and the preliminary theorems. Draw the figures carefully, using a ruler; then consider on what previous work the statements to be proved depend.

Theorem 1. If two triangles have two sides and the included angle of the one respectively equal to two sides and the included angle of the other, the triangles are congruent.

Given

To prove Proof. 1.

that A ABC=A A'B'C'.
Place A A'B'C' on ▲ ABC so that
A' falls on A, and

A'B' coincides with its equal AB.

2. Then A'C' may be caused to fall on AC,

5.

(State post. 1.)

Given

Given

Post. 1

.A ABC A A'B'C', by definition of congruence.

NOTES. This is a proof by superposition.

The theorem may be stated, A triangle is determined when two sides and the included angle are given.

EXERCISES. 31. If, in a quadrilateral ABCD,

ABDA, and diagonal AC bisects

that BC CD, and that AC bisects

=

BAD, prove

DCB.

32. Show that the distance BA across a lake may be measured by setting up a stake at O, sighting across it to fix the lines A'B and B'A, laying off OA' OA, and OB' OB, and then measuring B'A'.

=

B

Theorem 2. If two triangles have two angles and the included side of the one respectively equal to two angles and the included side of the other, the triangles are congruent.

Given

To prove
Proof. 1.

the ABC and A'B'C' such that

that

AB = A'B',

ZA=ZA',

B=B'.

AABC=1 A'B'C'.

Place A A'B'C' on ▲ ABC so that
A' falls on A, and

A'B' coincides with its equal AB.

2. Then A'C' may be caused to lie along AC,

4. .. C' falls on both AC and BC, and hence at their intersection.

5. .. ▲ ABCAA'B'C', by definition of congruence. NOTE. Th. 2, and th. 3 following, are attributed to Thales.

RECIPROCAL THEOREMS. The student will notice that theorems 1 and 2 have some similarity. Indeed, if the words side and angle be interchanged in th. 1 it becomes th. 2, and if interchanged in th. 2 it becomes th. 1. Theorems of this kind are called reciprocal. The relation is more clearly seen by resorting to parallel columns.

Th. 1. If two triangles have two sides and the included angle of the one respectively equal to two sides and the included angle of the other, the triangles are congruent.

Th. 2. If two triangles have two angles and the included side of the one respectively equal to two angles and the included side of the other, the triangles are congruent.

The principle involved is called the Principle of Reciprocity or of Duality, and is extensively used in geometry. But the student must not suppose that because a theorem is true its reciprocal theorem is also true; in elementary geometry, involving measurements, the reciprocal is often false. The principle is, however, of great value even here, for it leads. the student to see the relation between propositions, and it often suggests new possible theorems for investigation. For these purposes we shall use it.

At present it is sufficient to say that for many theorems of plane geometry reciprocal theorems may be formed by replacing the words

point by line,

line by point,

angles of a triangle by (opposite) sides of a triangle, sides of a triangle by (opposite) angles of a triangle.

The principle is still further illustrated by the fact that if two figures are congruent,

(a) Corresponding points lie on corresponding lines.

(b) To a line-segment between two points corresponds the linesegment between the corresponding points.

(a) Corresponding lines pass through corresponding points.

(b) To the intersection (or to the angle) of two lines corresponds the intersection (or the angle) of the corresponding lines.

EXERCISES. 33. Draw two congruent triangles and explain the above (a), (b), (a), (b) with reference to the figures.

34. Explain this statement and tell why it is true: Any two sides and the included angle of a triangle determine the remaining parts.

35. State the reciprocal of ex. 34 and tell whether it is true, and why.