1. If the areas of two triangles are not equal, then the two

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