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 || 1 ̧ and if ≤1 = 30°, find the other angles in the first

figure.

7. If 12, how are the bisectors of 21 and 3 related? Of ≤3

and 24? Of 21 and 22? Use § 102 for the last case.

8. If 12, and AO=OB, show that DO=OC.

State this theorem fully and prove it.

9. If 2 and 22 = 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.

90. THEOREM. If two lines cut by a í equal alternate interior angles, the lines

Given the lines ↳ and 11⁄2 cut by ʼn so that 21=22.

To prove that [72.

Proof: Suppose the lines, and l were to meet on the right of the transversal. Then a triangle would be formed of which ≤1 is an exterior an opposite interior angle.

This gives an exterior angle of a tria opposite interior angle, which is imposs Repeat this argument, supposing ↳ and left side of the transversal.

Hence ↳1⁄2 and ↳1⁄2 cannot meet and are pa

91. The type of proof used here is c proof. It consists in showing that some or contradictory results if the theorem is s

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

Given the lines ↳ and 11⁄2 cut by t so that / 2 = /3.

To prove that 1 || 2.

Proof : Quote the authority for each of the following steps:

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 l1 through P || to l.

Construction. Through P draw any line making a convenient angle, as ≤1 with 7.

Through P draw the line l1, making

22=21 (§ 47). Then | 7. ||

Proof Use the theorem, § 92.

Hereafter all constructions should be

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 equal to two right angles.

Given AABC with Z1, Z2, Z3. C

Hence, replacing 25 and 24 by their equals, <1 and 2, we have 21+2+3 = 2 rt .

HISTORICAL NOTE. This is one of the famous theorems of geometry. 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.

103. THEOREM. An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

The proof is left to the student. Compare this theorem with that of § 83.

104. Definition. A theorem which follows very easily from another theorem is called a corollary of that theorem. E.g. the theorem in § 103 is a corollary of that in § 102.

A

1. Find each angle of an equiangular triangle.

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B

2. If one angle of an equiangular triangle is bisected, find all the angles in the two triangles thus formed.

3. If in a ▲ ABC, AB'= AC' and ≤ A = ZB + ≤ C, find each angle. 4. If in the figure AB AC and ≤4 = 120°, find

21, 22, 23.

=

5. If one acute angle of a right triangle is 30°, what is the other acute angle? If one is 41° 23'?

6. If in two right triangles an acute angle of one is equal to an acute angle of the other, what can be

said of the remaining acute angles. What axiom is involved?

7. If in two right triangles the hypotenuse and an acute angle of one are equal respectively to the hypotenuse and an acute angle of the other, the triangles are congruent. Prove in full.

8. Can a triangle have two right angles? Two obtuse angles? Can the sum of two angles of a triangle be two right angles? What is the sum of the acute angles of a right triangle?

9. If two angles of a triangle are given how can the third be found? If the sum of two angles of one triangle is equal to the sum of two angles of another, how do the third angles compare?

10. Prove the theorem of § 102, using each of the figures in the margin. The first of these figures was used by Pythagoras.