1. Chapter 2

2. One of the basic axioms of Euclidean geometry says that two points determine a unique line. EXISTENCE AND UNIQUENESS

3. Two lines that don’t intersect are called parallel. This implies that two distinct lines cannot intersect in two or more points, they can either intersect in only one point or not at all.

4. PROBLEM Given a line and a point P not on ‚ construct a line through P and parallel to.

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6. let A be any point on, and draw Then draw a line so that as shown in the figure. This will be the desired line.

7. If and are not parallel, we may assume without loss of generality that they intersect as in the figure on the side of B at the point C. Now consider. the exterior angle is equal to the interior angle. But this contradicts the exterior angle theorem, which states that. The proof will be by contradiction. Hence must be parallel to.

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9. Given a line and a point P not on, there exists a line that contains P and is parallel to.

10. COROLLARY Given lines, then is parallel to. if as in the figure,and

11. THE PARALLEL POSTULATE If is any line and P is a point not on. parallel to. then there is no more than one linethrough P

12. Opposite Interior Angles Theorem are opposite interior angles. Let and be parallel lines with transversal such that and Then.

14. The proof will be by contradiction. then we could If the theorem and if construct a distinct line through P such that. are opposite interior angles,their congruence implies that. Since and was false,

15. and are two different lines, each goes But this is now a contradiction of the parallel postulate : So these angles must be congruent. we assumed that through P and each is parallel to. This contradiction comes about because.

17. THEOREM Let be any triangle. then.

19. Proof Let be the line through A parallel to such that anglesand and are opposite interior angles, as in figure. so Hence and are opposite interior and. =180. Since, and all together make a straight line.

20. Let be any quadrilateral. then

22. We draw the diagonal thus breaking the quadrilateral into two triangles.Note that +

23. The first sum of the last expression represents the sum of the angles of and the second sum represents the sum of the angles of. Hence, each is 180 and together they add up to 360.

24. COROLLARY(SAA) In and assume that and then..

25. Given a quadrilateral ABCD, the following are equivalent: and.. 3.The diagonals bisect each other.

26. LEMMA Let be a line. P a point not on. And A and B distinct points on such that is perpendicular to. Then.

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28. THEOREM Let and be parallel lines and let P and Q Then the distance from P to be points on equals the distance from Q to

29. Proof Draw lines from P and from Q perpendicular to, Meetingat B and at C, respectively. Since and, these angles are congruent,.Moreover is congruent to the supplement of.

31. So By opposite interior angles. must be a parallelogram, since opposite sides are Parallel. Hence Similarly,. Therefore, as claimed.