Show that these triangles are superposable; that so are all other triangles fulfilling the stated conditions; and that the solution is then unique.

III. Let the given angle A be obtuse.

(1) Let a be less than b, or a equal b. No solution.

Show that this conclusion could also be drawn from the fact that there cannot be two obtuse angles in a triangle (80). (2) Let a exceed b.

The points B, B' are on opposite sides of 4, and the triangle AB'C is not a solution.

a

b

α

Any other triangle having the specified parts equal to a, b, 4, would have the angle opposite b equal either to B or to its supplement (97). In the former case the triangle would be superposable on ABC. In the latter case the triangle could not exist since the supplement of B is obtuse, and there cannot be two obtuse angles in a triangle.

Hence there is but one type of triangle answering the requirements, and the solution is unique.

Ex. 1. Give a summary of the limitations there are on the data in order that any solution may be possible.

Ex. 2. Summarize the circumstances under which there is a unique solution.

Ex. 3. When is the solution ambiguous ? Quote the previous theorem from which it is inferred that there can be no third type of triangle fulfilling the conditions.

It has per

137. Determinate and indeterminate solutions. haps been noticed that the assigned conditions in these five problems of construction of a triangle correspond respec

tively to the five conditions of equality of two triangles (6466, 95, 97); and that the determinateness of the solution is in each case tested by applying the corresponding condition of equality.

For instance, the solution of 134 is uniquely determinate, because any two triangles that have two sides and the included angle in each respectively equal, are superposable by the first condition of equality.

Again, the solution of 136 is in one case ambiguously determinate, because any two triangles that have two sides and the angle opposite one of them in each equal have the angles opposite the other equal sides either equal or supplemental, and because there is (in the case referred to) no way of deciding the species of the angle in question from previous principles.

In each of these problems three parts were given. If only two parts are assigned, an indefinite number of distinct types of triangles answering the requirements can be constructed.

For instance, if only two sides are given, the included angle can be assumed at pleasure, and an indefinite number of distinct triangles can be formed so as to have the given sides.

Similarly if only two angles are given, the adjacent side can be assumed arbitrarily, and the solution is indeterminate.

The three angles, however, do not constitute three independent data; for then nothing more is given than when only two angles are assigned (as the third angle could be found by subtracting the sum of the first two from a straight angle).

Thus the problem to construct a triangle having its three angles equal to three assigned angles, is either impossible or indeterminate; impossible if the sum of the three given angles is not equal to a straight angle; indeterminate if it is.

EXERCISES

1. Given base, vertical angle, and sum of sides, construct the triangle.

Outline. In the figure of Art. 87, in which AD is the sum of the sides AC and CB, show that the angle ADB is half the vertical angle ACB; and hence that the triangle ADB can be constructed from the data (136). Then show how to construct the triangle ABC.

2. Given base, vertical angle, and difference of sides, construct the triangle.

Outline. In the figure of Art. 81, in which CD is the difference of sides, show that the angle CDB equals the sum of the vertical angle and half its supplement. Construct the triangle CDB by 136, and then the triangle ABC.

3. Given base, difference of sides, and difference of base angles, construct the triangle.

[Use 128, ex. 4.]

4. Given base, an adjacent angle, and the sum (or difference) of the other two sides, construct the triangle.

5. Given the angles and the perimeter, construct the triangle. Analysis. Prolong base BC both ways, making BB' equal BA, and CC' equal CA. Prove B'C' equal to perimeter; and angle B' equal half B, etc. Give synthesis.

6. Given the angles and the sum (or difference) of two sides, construct the triangle.

7. In the figure of Art. 81, prolong CA until AE equals AB, and draw EB; prove that angle DBE equals the sum of DEB and BDE, and is a right angle.

8. Construct a triangle, being given the base, the sum of sides, and the difference of the base angles. [Use ex. 7.]

9. Construct an equilateral triangle such that the perpendicular from the vertex to the base may be equal to a given line.

MCM. ELEM. GEOM.- 6

QUADRANGLES

Attention has hitherto been given to various properties of the plane figures formed by two or three straight lines. The figure that next presents itself is that formed by four lines each of which meets the next one in order.

138. Definitions. A plane figure formed by four line-segments that inclose a portion of the plane surface is called a quadrilateral figure, or a quadrangle.

These line-segments are called the sides, and their extremities the vertices of the quadrangle.

The angles formed by adjacent sides, and situated toward the interior of the boundary, are called the interior angles of the quadrangle, or briefly the angles.

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

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

If all of the conjunct angles are convex (21), the quadrangle is called convex.

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

In a convex quadrangle all the interior angles are concave, and no side when prolonged traverses the figure; but a concave quadrangle has one of the interior angles convex, and the sides of this angle traverse the figure when prolonged.

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

The sum of the sides is called the perimeter, and the sum of the angles the angle-sum.

In a convex quadrangle 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.

The four sides and four angles are called the eight parts of the quadrangle.

Primary relations of parts.

139. RELATION 1.

The sum of any three sides is

greater than the fourth (89).

140. Cor. The sum of any two sides is greater than the difference of the other two.

141. RELATION 2. The angle-sum is equal to a perigon.

[Divide the quadrangle into two triangles by a diagonal and apply 129.]

142. Cor. I. A conjunct angle is equal to the sum of the three non-adjacent interior angles.

143. Cor. 2. Only one of the interior angles in a quadrangle can be convex.

144. Cor. 3. If two quadrangles have three angles of one equal respectively to three angles of the other, the remaining angles are equal.

Ex. 1. The sum of the four sides of a quadrangle is greater than the double of either diagonal, and greater than the sum of the diagonals.

Ex. 2. The sum of the four interior angles is equal to one third of the sum of the four conjunct angles.