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parallel lines

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

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parallel lines

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  1. Chapter 2 parallel lines

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

  3. 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. Two linesthat don’t intersect are called parallel.

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

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  6. Solution 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. Proof The proof will be by contradiction. Ifand 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 tothe interior angle . But this contradicts the exterior angletheorem, which states that . Hence must be parallel to .

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

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

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

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

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

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

  15. APPLICATION

  16. THEOREM Let be any triangle. then .

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

  18. COROLLARY Let be any quadrilateral. then

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

  20. 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.

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

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

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

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

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

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

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