Construct in succession DA, DAB, AB, ABM respectively equal to the given parts. With radius equal to c and center D, describe an arc cutting BM in C, C'.

State when there is no solution, when two solutions, when only one.

Given four sides and an angle.

186. PROBLEM 19. To construct a quadrangle having four sides equal to four given lines and one angle equal to a given angle.

Let a, b, c, d be the given lines, and 4 the given angle.

Let it be required to construct a quadrangle whose sides taken in order may be equal to a, b, c, d, and such that the sides a and d may contain an angle equal to 4.

Construct the triangle ABD having AB equal to a, AD equal to d, and the included angle DAB equal to ▲ (133). Next on BD construct the triangle BCD having BC equal to b, and CD equal to c (132).

Show by suitable figures that there may be two solutions, only one solution, or no solution.

NOTE. The case in which three sides and two opposite angles are given is postponed (III, 198).

Ex. 1. Compare the data in the three unique solutions above with the three conditions of equality (145-147).

Ex. 2. Construct a trapezoid being given two adjacent sides, the included angle, and the angle opposite to the latter. Which case does this come under? Discuss the solution.

EXERCISES

1. Draw a line such that its segment intercepted between two given fixed indefinite lines shall be equal and parallel to a given finite line.

2. Draw a line parallel to the base of a triangle, cutting the sides so that the sum of the two segments adjacent to the base shall be equal to a given line.

Analysis. Let PB, QC be the two segments. Draw PD parallel to QC. Then in the triangle BPD, the angles and the sum of two sides are given (137, ex. 6). Give synthesis.

3. Construct a parallelogram being given two adjacent sides and a diagonal.

4. Construct a parallelogram being given a side and two diagonals. 5. Inscribe a rhombus in a triangle having one of its angles coincident with an angle of the triangle.

6. One angle of a parallelogram is given in position and the point of intersection of the diagonals is given; construct the parallelogram.

7. If the diagonals of a quadrangle bisect its angles, then it is a rhombus.

8. The perimeter of a quadrangle is greater than the sum of its diagonals.

9. The sum of two sides of a triangle is greater than double the median drawn to the third side; and the perimeter of the triangle is greater than the sum of the three medians.

10. Given two medians and their included angle, construct the triangle.

POLYGONS

This section considers the figure formed by any number of lines each of which meets the next in order, and generalizes some of the results obtained in the preceding sections.

187. Definitions. A plane figure composed of segments of straight lines that inclose a portion of the plane surface, is called a polygon.

These segments are called the sides, their extremities the vertices, and their sum the perimeter, of the polygon.

A line joining any two non-adjacent vertices is called a diagonal.

The angles formed by consecutive sides, and situated towards the interior of the boundary, are called the interior angles of the polygon.

The exterior angles conjunct to these will be called for brevity the conjunct angles.

If all of the conjunct angles are convex, the polygon is called a convex polygon.

If one of the conjunct angles is concave, the polygon is said to be concave at that angle.

In a convex polygon each of the interior angles is concave, and its exterior conjunct angle is convex.

A concave polygon has at least one of the conjunct angles concave, and the corresponding interior angle convex. The sides

of this angle traverse the figure if prolonged.

In any polygon the concave angle formed by one side and the prolongation of an adjacent side is called an exterior angle of the polygon.

A polygon whose sides are all equal is equilateral, and one whose angles are all equal is equiangular.

A polygon which is both equilateral and equiangular is regular.

Two polygons that have the sides of one respectively equal to the sides of the other, taken in order, are said to be mutually equilateral, or one is said to be equilateral to the other.

Two polygons that have the angles of one respectively equal to the angles of the other, taken in order, are said to be mutually equiangular, or one is said to be equiangular to the other.

In two mutually equiangular polygons the vertices of equal angles are said to correspond; and the sides joining corresponding vertices are called corresponding sides.

The two polygons are said to be directly equiangular if the sides of two corresponding angles can be brought into coincidence in such a way that their corresponding sides may coincide, without turning either polygon out of the plane; otherwise the polygons are said to be obversely equiangular to each other.

Two equal polygons are said to be directly superposable when they can be superposed without turning either polygon out of its plane; otherwise the equal polygons are said to be obversely superposable.

The number of interior angles in a polygon is equal to the number of sides.

A polygon of five sides is called a pentagon, of six sides a hexagon, of seven sides a heptagon, of eight sides an octagon, of nine sides a nonagon, of ten sides a decagon. A twelve-sided polygon is called a dodecagon, and a fifteen-sided one a pentadecagon. In the discussion of general properties, a polygon of n sides is called an n-gon.

The sum of the n interior angles is called the interior angle-sum.

In a convex polygon, the sum of the exterior angles formed by prolonging each side one way, no two adjacent sides being prolonged through the same vertex, is called the exterior angle-sum.

GENERAL PROPERTIES OF POLYGONS

The following preliminary general theorems will be of frequent use in the theory of the polygon.

Division into triangles by diagonals.

188. THEOREM 39. In any n-gon if all possible diagonals are drawn in any manner, except that no two intersect within the polygon, then there will be n-3 such diagonals, and the n-gon will be divided inton-2 triangles.*

B

Let the diagonals be drawn as stated. Begin with at diagonal, such as AC, that joins two alternate vertices, and call this the first diagonal. This first diagonal cuts off one triangle from the n-gon and leaves an (n-1)-gon. Similarly some second diagonal cuts off a second triangle from this and leaves an (n − 2)-gon. A third diagonal cuts off a third triangle from the latter and leaves an (n-3)-gon, and so on. When n-3 diagonals are so drawn, there are n-3 triangles cut off, and there is left an [n-(n-3)]-gon, that is a 3-gon, or triangle. Thus there are n 3 diagonals and n triangles. Hence the n-gon is divided into n - 2 triangles.

3+1

NOTE. The student who may not be familiar with algebraic symbols may apply this method of reasoning to the special case of the hexagon or heptagon.

Another mode of proof consists in beginning with a single triangle, and then adding other triangles, so as to form in succession a quadrangle, a pentagon, a hexagon, etc.

*The symbol n-2 is read n minus 2, and stands for the number which is 2 units less than n.