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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 = 90º 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 = 90º 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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