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.
- 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.
|1.||l m cut by a transversal t, 1 ⊥ t||Given|
|2.||∠1 is right||Definition of perpendicular|
|3.||m∠1 = 90º||Definition of right angle|
|4.||∠1 and ∠2 are corresponding angles||Definition of corresponding angles|
|5.||∠4 ~= ∠8||Postulate 10.1|
|6.||m∠1 = m∠2||Definition of ~=|
|7.||m∠2 = 90º||Substitution (steps 2 and 5)|
|8.||∠2 is right||Definition of right angle|
|9.||m ⊥ t||Definition of ⊥|
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.