Another test of parallelism. Arts. 123 and 124 furnish a useful test as to whether two lines are parallel or not (compare 119). Art. 124 is often used to prove that certain lines will meet if prolonged in a certain way.

125. Cor. 5. Lines that are parallel to the same line are parallel to each other. (Use 122.)

Ex. A line perpendicular to one of two parallels is perpendicular to the other.

Angles having parallel sides.

126. THEOREM 23. If two angles have the two sides of one respectively parallel to the two sides of the other (parallel lines being at the same side of the line joining the vertices of the angles), then the angles are equal.

Let AOB, A'O'B' be the angles having OA and O'A' parallel and on the same side of 00'; and similarly for OB and O'B'.

B

Β'

A

To

prove that the angles AOB and A'O'B' are equal. Let O'A' meet OB (extended if necessary) in C.

The angles AOB, A'O'B' are each equal to A'CB (122).
Hence, they are equal to each other.

127 (a). Cor. I. If two angles have the two sides of one respectively parallel to the two sides of the other (parallel lines being at opposite sides of the line joining the vertices of the angles), then the angles are equal.

127 (b). Cor. 2. If two angles have the two sides of one respectively parallel to the two sides of the other (two of the parallels being at the same side of the line joining the vertices and the other two being on opposite sides of that line), then the angles are supplemental.

THEORY OF PARALLELS APPLIED TO ANGLE-SUMS

The following two theorems, with their inferences, illustrate how the theory of parallels may be used in the addition and subtraction of angles.

Sum of two angles of a triangle.

128. THEOREM 24. When any side of a triangle is extended, the exterior angle is equal to the sum of the two interior opposite angles.

Let the side AC of the triangle ABC be extended to E.

To prove that the exterior angle BCE is equal to the sum of the opposite interior angles CAB and CBA.

Draw the line CD parallel to AB (114).

The angles ECD and CAB are equal (122).

Also the angles DCB and CBA are equal (117).

Therefore, by addition of equals, the angle ECB is equal

to the sum of the angles CAB and CBA.

Ex. 1. When two lines are met by a transversal, the difference of two corresponding angles is equal to the angle between the two lines. Ex. 2. The difference between two alternate angles is equal to what angle?

By how much does the sum of the two interior angles at one side of a transversal exceed the sum of the interior angles at the other side? Ex. 3. In the figure of Art. 87, prove that the angle ADB equals half ACB.

Ex. 4. In the figure of Art. 81, prove that the difference of the angles ABC and ACB equals double DBC.

[Angle ABC equals the sum of ABD and DBC, which equals the sum of ADB and DBC, etc.]

Angle-sum in a triangle.

129. THEOREM 25. The sum of the three interior angles of a triangle is equal to a straight angle.

[Use the equality proved in 128; and add to both members the third interior angle.]

Ex. 1. In a right triangle the acute angles are complemental; and in an isosceles right triangle the acute angles are each equal to half a right angle.

Ex. 2. In any isosceles triangle each of the equal angles is equal to the complement of half the third angle.

Ex. 3. In an equilateral triangle each angle is equal to two thirds of a right angle.

Ex. 4. Show how to construct an angle equal to one third of a right angle. Hence show how to trisect a given right angle.

Ex. 5. Trisect a given straight angle.

130. Cor. If two triangles have two angles of one equal to two angles of the other, then the third angles are equal.

EXERCISES

1. Through a given point draw a line making with a given line an angle equal to a given angle.

2. If two lines are respectively perpendicular to two other lines, the angles formed by the first pair are respectively equal to the angles formed by the second pair.

3. Two lines perpendicular to two parallel lines, respectively, are parallel.

CONSTRUCTION OF TRIANGLES

131. There is a large class of problems involving the construction of triangles to satisfy certain prescribed conditions involving the sides and angles.

In a triangle the three sides and the three angles are called its six parts. For convenience the sides will be denoted by a, b, c and the opposite angles respectively by A, B, C. In a right triangle the side opposite the right angle is called the hypotenuse.

The simplest condition that can be imposed on the construction of a triangle is the assignment of certain line segments or angles to which some of the six parts are to be made equal. Those parts which are to be made equal to prescribed segments or angles are said to be given or known, and the remaining parts, about which nothing is prescribed, are said to be unknown.

In each of the five following problems three of the six parts are given, and it is required to find by a geometric construction a triangle answering to the prescribed conditions, and incidentally to determine the three unknown parts of the triangle.

The solution of such a problem has three divisions:

(1) To make the actual construction by means of processes that ultimately involve only the drawing of straight lines and of circular arcs.

(2) To prove by the use of previous propositions that the figure so constructed satisfies the prescribed conditions.

(3) To discuss the solution; i.e. to examine what limitations there are on the data so that it may be possible to satisfy the demands; and under what circumstances it will be possible to satisfy them in only one way, or in more than one way; and also to examine certain special and limiting cases.

When the data are such that the demands cannot be

satisfied by any triangle, the problem is said to have no solution.

If they can be satisfied by one, and only one, type of triangle (i.e. if all the triangles fulfilling the stated conditions can be superposed), the problem is said to have a unique solution.

If the demands can be satisfied by two or some greater definite number of distinct types of triangles not capable of superposition, the problem is said to have a determinate but ambiguous solution.

If the demands can be satisfied by an indefinite number of triangles, the problem is said to have an indeterminate solution.

To prepare the way for the solution it is usually best to make a preliminary analysis of each problem. This may be described in a general way as follows:

Suppose the problem solved, and the required figure drawn. Mark the parts that are supposed to be equal to "given" lines or angles. Analyze the figure to discover the relations of the known and unknown parts. Draw any

lines that may help to bring them into closer relations to each other. Observe which of the problems already solved could be used to construct the various parts subject to the given conditions. This is called "reducing the problem to previous ones."

After this reduction (or analysis) has been made, perform these simpler constructions in their proper order, thus building up the figure step by step. Such building-up process is called a synthesis. Then will follow the proof, and the discussion as already stated.

In many cases the preliminary analysis is so simple that it will not be given, but the student should always make such analysis before consulting the synthetic solution.

It is advisable to make some of the actual constructions with ruler and compasses; and it affords better geometrical training to dispense with all other mechanical aids.