Geometry: Using Parallelism to Prove Perpendicularity

Using Parallelism to Prove Perpendicularity

Suppose you have the situation shown in Figure 10.7. Two lines, l and m, are parallel, and are cut by a transversal t. In addition, suppose that 1 ? t. In this case, you can conclude that m ? t. There are those who would doubt your conclusions, and it is for those people that I include a proof. As it is stated, the problem cannot have theorem status. Theorems are typically general statements, like ?when two lines intersect, the vertical angles formed are congruent.? In this case, your observation came from a specific situation, and it cannot become a theorem unless it is written in more general terms, like ?when two parallel lines are cut by a transversal, if one of the lines is perpendicular to the transversal, then both of the lines are perpendicular to the transversal.? That's the stuff that theorems are made of. Here's a formal proof of the theorem.

Figure 10.7Lines l and m are parallel lines cut by a transversal t, with 1 ? t.

  • Theorem 10.6: When two parallel lines are cut by a transversal, if one of the lines is perpendicular to the transversal, then both of the lines are perpendicular to the transversal.

Figure 10.7 illustrates the situation nicely.

  • Given: Lines l and m are parallel and are cut by a transversal t, 1 ? t.
  • Prove: m ? t
  • Proof: Your game plan is to use Postulate 10.1, which says that when two parallel lines are cut by a transversal, corresponding angles are congruent. Because l and t meet to form a right angle, so will m and t, making them perpendicular.
StatementsReasons
1.l ? ? m cut by a transversal t, 1 ? tGiven
2. ?1 is rightDefinition of perpendicular
3. m?1 = 90 Definition of right angle
4. ?1 and ?2 are corresponding anglesDefinition of corresponding angles
5. ?4 ~= ?8 Postulate 10.1
6. m?1 = m?2Definition of ~=
7. m?2 = 90 Substitution (steps 2 and 5)
8. ?2 is rightDefinition of right angle
9. m ? t Definition of ?
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Excerpted from The Complete Idiot's Guide to Geometry 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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