# 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.

Statements | Reasons | |
---|---|---|

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.

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