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Geometry: Proving Lines Are Parallel

Proving Lines Are Parallel

When you were given Postulate 10.1, you were able to prove several angle relationships that developed when two parallel lines were cut by a transversal. There are times when particular angle relationships are given to you, and you need to determine whether or not the lines are parallel. You'll develop some theorems to help you do this easily. Your first theorem, Theorem 10.7, will be established using contradiction. The rest of the theorems will follow using a direct proof and Theorem 10.7.

Figure 10.8l and m are cut by a transversal t, ∠1 and ∠2 are corresponding angles.

Let's review the steps involved in constructing a proof by contradiction. Start by assuming that the conclusion is false, and then showing that the hypotheses must also be false. In the original statement of the proof, you start with congruent corresponding angles and conclude that the two lines are parallel. To prove this theorem using contradiction, assume that the two lines are not parallel, and show that the corresponding angles cannot be congruent.

Figure 10.9l and m are cut by a transversal t, l ‌ ‌ m, r ‌ ‌ l, and r, m, and l intersect at O.

  • Theorem 10.7: If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel.

A drawing of this situation is shown in Figure 10.8. Two lines, l and m are cut by a transversal t, and ∠1 and ∠2 are corresponding angles.

  • Given: l and m are cut by a transversal t, l ‌/‌ m.
  • Prove: ∠1 and ∠2 are not congruent (∠1 ~/= ∠2).
  • Proof: Assume that l ‌/‌ m. Because l and m are cut by a transversal t, m and t must intersect. You might call the point of intersection of m and t the point O. Because l is not parallel to m, we can find a line, say r, that passes through O and is parallel to l. I've drawn this new line in Figure 10.9. In this new drawing, ∠3 and ∠2 are corresponding angles, so by Postulate 10.1, they are congruent. But wait a minute! If ∠2 ~= ∠3, and m∠3 + m∠4 = m∠1 by the Angle Addition Postulate, m∠2 + m∠4 = m∠1. Because m∠4 > 0 (by the Protractor Postulate), this means that m∠2 < m∠1, and ∠1 ~/= ∠2. Let's put this all down in two columns.
 StatementsReasons
1. l and m are two lines cut by a transversal t, with 1 |/| mGiven
2. Let r be a line passing through O which is parallel to lEuclid's 5th postulate
3. ∠3 and ∠2 are corresponding anglesDefinition of corresponding angles
4. ∠2 ~= ∠3 Postulate 10.1
5. m∠2 = m∠3 Definition of ~=
6. m∠3 + m∠4 = m∠1 Angle Addition Postulate
7. m∠2 + m∠4 = m∠1 Substitution (steps 5 and 6)
8. m∠4 > 0 Protractor Postulate
9. m∠2 < m∠1 Definition of inequality
10. ∠4 ~/= ∠8 Definition of ~=

That completes your proof by contradiction. The rest of the theorems that you prove in this section will make use of Theorem 10.7. The rest of the theorems in this section are converses of theorems proved earlier.

Let's take a look at some other angle relationships that can be used to prove that two lines are parallel. These two theorems are similar, and to be fair I will prove the first one and leave you to prove the second.

  • Theorem 10.8: If two lines are cut by a transversal so that the alternate interior angles are congruent, then these lines are parallel.
  • Theorem 10.9: If two lines are cut by a transversal so that alternate exterior angles are congruent, then these lines are parallel.

Figure 10.10 shows two lines cut by a transversal t, with alternate interior angles labeled ∠1 and ∠2.

Figure 10.10l and m are cut by a transversal t, and ∠1 and ∠2 are alternate interior angles.

  • Given: l and m are cut by a transversal t, with ∠4 ~= ∠8.
  • Prove: l ‌ ‌ m.
  • Proof: The game plan is simple. In order to use Theorem 10.7, you need to show that corresponding angles are congruent. You can use the fact that ∠1 and ∠2 are vertical angles, so they are congruent. Because ∠2 and ∠3 are corresponding angles, if you can show that they are congruent, then you will be able to conclude that your lines are parallel. The transitive property of congruence will put the nail in the coffin, so to speak.
 StatementsReasons
1. l and m are two lines cut by a transversal t, with ∠4 ~= ∠8 Given
2. ∠1 and ∠3 are vertical anglesDefinition of vertical angles
3. ∠1 ~= ∠3 Theorem 8.1
4. ∠2 and ∠3 are corresponding anglesDefinition of corresponding angles
5. ∠2 ~= ∠3 Transitive property of ~=
6. l ‌ ‌ mTheorem 10.7

Theorem 10.4 established the fact that if two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary angles. Theorem 10.5 claimed that if two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary angles. It's now time to prove the converse of these statements. Let's split the work: I'll prove Theorem 10.10 and you'll take care of Theorem 10.11.

  • Theorem 10.10: If two lines are cut by a transversal so that the interior angles on the same side of the transversal are supplementary, then these lines are parallel.
  • Theorem 10.11: If two lines are cut by a transversal so that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel.

Figure 10.11 will help you visualize this situation. Two lines, l and m, are cut by a transversal t, with interior angles on the same side of the transversal labeled ∠1 and ∠2.

Figure 10.11l and m, are cut by a transversal t, and ∠1 and ∠2 are interior angles on the same side of the transversal.

  • Given: l and m are cut by a transversal t, ∠1 and ∠2 are supplementary angles.
  • Prove: l ‌ ‌ m.
  • Proof: Here's the game plan. In order to use Theorem 10.7, you need to show that corresponding angles are congruent. But it might be easier to use Theorem 10.8 if you can show that ∠2 and ∠3 are congruent. You can do that fairly easily, if you apply what you discovered. Because ∠1 and ∠3 are supplementary angles, and ∠1 and ∠2 are supplementary angles, you can conclude that ∠2 ~= ∠3. Then you apply Theorem 10.8 and your work is done.
 StatementsReasons
1.l and m are two lines cut by a transversal t, ∠1 and ∠2 are supplementary angles.Given
2.∠1 and ∠3 are supplementary anglesDefinition of supplementary angles
3.∠2 ~= ∠3∠1 and ∠3 are supplementary angles, and ∠1 and ∠2 are supplementary angles
4.l ‌ ‌ mTheorem 10.8

In a complicated world, a complicated theorem requires a complicated drawing. If your drawing is too involved, it could be difficult to decide which lines are parallel because of congruent angles. Consider Figure 10.12. Suppose that ∠1 ~= ∠3. Which lines must be parallel? Because ∠1 and ∠3 are corresponding angles when viewing lines o and n cut by transversal m, o ‌ ‌ n.

Figure 10.12The intersection of lines l, m, n, and o.

Put Me in, Coach!

Here's your chance to shine. Remember that I am with you in spirit and have provided the answers to these questions in Answer Key.

  1. If l ‌ ‌ m as in Figure 10.4, with m∠2 = 2x - 45 and m∠1 = x, find m∠6 and m∠8.
  2. Write a formal proof for Theorem 10.3.
  3. Write a formal proof for Theorem 10.5.
  4. Prove Theorem 10.9.
  5. Prove Theorem 10.11.
  6. In Figure 10.12, which lines must be parallel if ∠3 ~= ∠11 ?
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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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