For each statement in Exercise 9 that can be proved by the

Proving Lines Parallel Proving Lines Parallel 2.3 KEY CONCEPTS Here is a quick review of the relevant postulate and theorems from Section 2.1. Each has the hypothesis “If two parallel lines are cut by a transversal.” If two parallel lines are cut by a transversal, then the corresponding angles are congruent. POSTULATE 11 If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. THEOREM 2.1.2 If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. THEOREM 2.1.3 If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. THEOREM 2.1.4 If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary. THEOREM 2.1.5 Suppose that we wish to prove that two lines are parallel rather than to establish an angle relationship as the previous statements do. Such a theorem would take the form “If . . . , then these lines are parallel.” At present, the only method we have of proving lines parallel is based on the definition of parallel lines. Establishing the conditions of the definition that coplanar lines do not intersect is virtually impossible Thus, we begin to develop methods for proving that lines in a plane are parallel by proving Theorem 2.3.1 by the indirect method. Counterparts of Theorems 2.1.2–2.1.5, namely, Theorems 2.3.2–2.3.5, are proved directly but depend on Theorem 2.3.1. Except for Theorem 2.3.6, the theorems of this section require coplanar lines. If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel. THEOREM 2.3.1 Exs. 1, 2 GIVEN: 艎 and m cut by transversal t See Figure 2.16 PROVE: 7 m ∠1 ⬵ ∠2 PROOF: Suppose that . Then a line r can be drawn through point P that is parallel to m; this follows from the Parallel Postulate. If , then because these angles correspond. But by hypothesis. Now by the Transitive Property of Congruence; therefore, . But . See Figure 2.16. Substitution of for leads to ; and by subtraction, . This contradicts the Protractor Postulate, which states that the measure of any angle must be a positive number. Then r and 艎 must coincide, and it follows that . Once proved, Theorem 2.3.1 opens the doors to a host of other methods for prov- ing that lines are parallel. Each claim in Theorems 2.3.2–2.3.5 is the converse of its counterpart in Section 2.1. 7 m m ∠4 = 0 m ∠1 + m∠4 = m∠1 m ∠3 m ∠1 m ∠3 + m∠4 = m∠1 m ∠3 = m∠1 ∠3 ⬵ ∠1 ∠1 ⬵ ∠2 ∠3 ⬵ ∠2 r 7 m 7 m r 1 2 4 3 t P m Figure 2.16 If two lines are cut by a transversal so that the alternate interior angles are congruent, then these lines are parallel. THEOREM 2.3.2 GIVEN: Lines 艎 and m and transversal t See Figure 2.17 PROVE: PLAN FOR THE PROOF: Show that corresponding angles. Then apply Theorem 2.3.1, in which corresponding imply parallel lines. ∠s ⬵ ∠1 ⬵ ∠2 7 m ∠2 ⬵ ∠3 PROOF Statements Reasons 1. 艎 and m; trans. t; 2.

3. 4.

7 m ∠1 ⬵ ∠2 ∠1 ⬵ ∠3 ∠2 ⬵ ∠3 1. Given 2. If two lines intersect, vertical are 3. Transitive Property of Congruence 4. If two lines are cut by a transversal so that corr. are then these lines are parallel ⬵, ∠s ⬵ ∠s The following theorem is proved in a manner much like the proof of Theorem 2.3.2. The proof is left as an exercise. If two lines are cut by a transversal so that the alternate exterior angles are congruent, then these lines are parallel. THEOREM 2.3.3 1 2 3 t m Figure 2.17 Discover When a staircase is designed, “stringers” are cut for each side of the stairs as shown. How are angles 1 and 3 related? How are angles 1 and 2 related? ANSWERS Congruent; Complementary In a more involved drawing, it may be difficult to decide which lines are parallel because of congruent angles. Consider Figure 2.18 on page 88. Suppose that . Which lines must be parallel? The resulting confusion it appears that a may be paral- lel to b and c may be parallel to d can be overcome by asking, “Which lines help form and ?” In this case, and are formed by lines a and b with c as the trans- versal. Thus, . a 7 b ∠3 ∠1 ∠3 ∠1 ∠1 ⬵ ∠3 1 2 3 쮿 Congruent, Complementary