90. THEOREM. If two lines cut by a transversal have equal alternate interior angles, the lines are parallel.

Given the lines 1 and 2 cut by t so that

21=22.

To prove that 1 || 72.

Proof: Suppose the lines l1 and l1⁄2 were to meet on the right of the trans

versal. Then a triangle would be

-11

-12

formed of which 1 is an exterior angle and 2 an opposite interior angle.

This gives an exterior angle of a triangle equal to an opposite interior angle, which is impossible. (Why?) Repeat this argument, supposing 1 and 2 to meet on the left side of the transversal.

Hence 1 and 2 cannot meet and are parallel (§ 89).

91. The type of proof used here is called an indirect proof. It consists in showing that something impossible or contradictory results if the theorem is supposed not true.

92. THEOREM. If two lines cut by a transversal have equal corresponding angles, the lines are parallel.

Given the lines ↳ and l1⁄2 cut by t so that

93. THEOREM. If two lines cut by a transversal have the sum of the interior angles on one side of the transversal equal to two right angles, the lines are parallel.

Given 1, and 1, cut by t so that ≤ 4 + ≤ 2 = 2 rt. .

To prove that l || 72.

Proof: 24 is supplementary to 21 and also to 2.

1. Show that if each of two lines is perpendicular to the same line, they are parallel to each other.

2. Let ABC be any triangle. Bisect BC at D. Draw AD and prolong it to make DE = AD. Draw CE. Prove CE || AB.

3. Use Ex. 2 to construct a line through a A given point parallel to a given line.

B

SUGGESTION. Let AB be a segment of the given line and let C be the given point. Draw CA and CB and proceed as in Ex. 2.

95. Exs. 2 and 3 above show that through a point P, not on a line 7, at least one line l1 can be drawn parallel to 7.

P

It seems reasonable to suppose that no other line l2 can be drawn through P parallel to 7, although this cannot be proved from the preceding theorems. See § 60.

Hence we assume the following:

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96. Axiom VIII. Through a point not on a line only one straight line can be drawn parallel to that line.

HISTORICAL NOTE. This so-called axiom of parallels has attracted more attention than any other proposition in geometry. Until the year 1829 persistent attempts were made by the world's most eminent mathematicians to prove it by means of the other axioms of geometry. In that year, however, a Russian, Lobachevsky, showed this to be impossible and hence it must forever remain an axiom unless some other equivalent proposition is assumed.

97. THEOREM. If two parallel lines are cut by a transversal, the alternate interior angles are equal.

But by hypothesis 2 || 1 and thus we have through P two lines parallel to l1, which is contrary to Ax. VIII. Therefore, the supposition that 2 is not equal to 21 leads to a contradiction, and hence 1= 2.

98. Compare this theorem with that of § 90. The hypothesis of either is seen to be the conclusion of the other.

When two theorems are thus related, each is said to be the converse of the other. Other pairs of converse theorems thus far are those in § 37 and Ex. 4, § 87; §§ 75 and 76, and §§ 84 and 85.

The converse of a theorem is never to be taken for granted without proof, since it does not follow that a statement is true because its converse is true.

Thus, it is true that if a triangle is equilateral, it is also isosceles, but the converse, if a triangle is isosceles, it is also equilateral, is not true.

99. THEOREM. If two parallel lines are cut by a transversal, the sum of the interior angles on one side of the transversal is two right angles.

Suggestion. Make use of the preceding theorem and give the proof in full. Of what theorem is this the converse?

1. State and prove the converse of the theorem in § 92.

2. Prove that if two parallel lines are cut by a transversal the alternate exterior angles are equal. Draw the figure.

3. State and prove the converse of the theorem in Ex. 2.

4. If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other also.

5. Two straight lines in the same plane parallel to a third line are parallel to each other. Suppose they meet and then use § 96.

6. If 11 || 12 || 7, and if ≤1 = 30°, find the other angles in the first figure.

7. If 7, how are the bisectors of 21 and 23 related? Of ≤3 and <4?

8. If 1, 2, and AO=OB, show that DO=OC.

State this theorem fully and prove it.

9. If, and ≤2 = 521, find 24 and 23.

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10. If two parallel lines are cut by a transversal, the sum of the exterior angles on one side of the transversal is two right angles.

11. State and prove the converse of the preceding theorem.

APPLICATIONS OF THEOREMS ON PARALLELS.

101. PROBLEM. Through a given point to construct a line parallel to a given line.

Given the line 7 and the point P outside of it.

To construct a line 7, through P || to l.

Construction. Through P draw any line making a convenient angle, as 21 with 7.

Through P draw the line l1, making

22=21 (§ 47). Then l || 7.

Proof: Use the theorem, § 92.

Hereafter all constructions should be

P/2

described fully as above, followed by a proof that the construction gives the required figure.

102. THEOREM. The sum of the angles of a triangle is

Hence, replacing ≤5 and 24 by their equals, 1 and 42, we have 21+22+23 = 2 rt .

HISTORICAL NOTE. This is one of the famous theorems of geoinetry. It was known by Pythagoras (500 B.C.), but special cases were known much earlier. The figure used here is the one given by Aristotle and Euclid. As is apparent, the proof depends upon the theorem, § 97, and thus indirectly upon Axiom VIII. The interdependence of these two propositions has been studied extensively during the last two centuries.