In a plane, if two lines are intersected by a transversal so

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 Theorems 2.3.4 and 2.3.5 enable us to prove that lines are parallel when certain pairs of angles are supplementary. EXAMPLE 1 In Figure 2.18, which lines must be parallel if Solution and are the alternate exterior angles formed when lines c and d are cut by transversal b. Thus, . 쮿 c 7 d ∠8 ∠3 ∠3 ⬵ ∠8? 2 4

6 1

3 5 8 7 a b c d Figure 2.18 EXAMPLE 2 In Figure 2.18, . Find such that . Solution With b as a transversal for lines c and d, and are corresponding angles. Then c would be parallel to d if and were congruent. Thus, . 쮿 m ∠5 = 94° ∠5 ∠3 ∠5 ∠3 c 7 d m ∠5 m ∠3 = 94° 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 2.3.4 EXAMPLE 3 Prove Theorem 2.3.4. See Figure 2.19. GIVEN: Lines 艎 and m; transversal t is supplementary to PROVE: 7 m ∠2 ∠1 The proof of Theorem 2.3.5 is similar to that of Theorem 2.3.4. The proof is left as an exercise. PROOF Statements Reasons 1. 艎 and m; trans. t; is supp. to 2. is supp. to

3. 4.

7 m ∠2 ⬵ ∠3 ∠3 ∠1 ∠2 ∠1 1. Given 2. If the exterior sides of two adjacent form a straight line, these are supplementary 3. If two are supp. to the same , they are 4. If two lines are cut by a transversal so that corr. are , then these lines are parallel ⬵ ∠s ⬵ ∠ ∠s ∠s ∠s 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. THEOREM 2.3.5 m t 1 2 3 Figure 2.19 쮿 We include two final theorems that provide additional means of proving that lines are parallel. The proof of Theorem 2.3.6 see Exercise 33 requires an auxiliary line a transversal. Proof of Theorem 2.3.7 is found in Example 5. EXAMPLE 4 In Figure 2.20, which line segments must be parallel if and are supplementary? Solution Again, the solution lies in the question “Which line segments form and ” With as a transversal, and are formed by and . Because B and C are supplementary, it follows that . 쮿 AB 7 DC ∠s DC AB ∠C ∠B BC ∠C? ∠B ∠C ∠B A D C B Figure 2.20 If two lines are each parallel to a third line, then these lines are parallel to each other. THEOREM 2.3.6 Theorem 2.3.6 is true even if the three lines described are not coplanar. In Theorem 2.3.7, the lines must be coplanar. If two coplanar lines are each perpendicular to a third line, then these lines are parallel to each other. THEOREM 2.3.7 General Rule: The proof of Theorem 2.3.7 depends upon establishing the condition found in one of the Theorems 2.3.1–2.3.6. Illustration: In Example 5, we establish congruent corresponding angles in statement 3 so that lines are parallel by Theorem 2.3.1. STRATEGY FOR PROOF 왘 Proving That Lines are Parallel Exs. 3–8 EXAMPLE 5 GIVEN: and See Figure 2.21 PROVE: Í AC ⬜ Í DF Í DF ⬜ Í BE Í AC ⬜ Í BE PROOF Statements Reasons

1. and

2. 1 and 2 are rt.

3. 4.

Í AC 7 Í DF ∠1 ⬵ ∠2 ∠s ∠s Í DF ⬜ Í BE Í AC ⬜ Í BE 1. Given 2. If two lines are perpendicular, they meet to form right 3. All right angles are 4. If two lines are cut by a transversal so that corr. are , then these lines are parallel ⬵ ∠s ⬵ ∠s 1 2 B A C E D F Figure 2.21 쮿 Construction 7 depends on Theorem 2.3.1, which is restated below. m t 1 2 3 Figure 2.22 EXAMPLE 6 GIVEN: and See Figure 2.22. FIND: x , so that 艎 will be parallel to m Solution For 艎 to be parallel to m, 1 and 2 would have to be supplementary. This follows from Theorem 2.3.4 because 1 and 2 are interior angles on the same side of transversal t. Then NOTE: With and , we see that and are supplementary. Then . 쮿 7 m ∠2 ∠1 m ∠2 = 75° m ∠1 = 105° x = 15 12x = 180 7x + 5x = 180 ∠s ∠s m ∠2 = 5x m ∠1 = 7x If two lines are cut by a transversal so that corresponding angles are congruent, then these lines are parallel. THEOREM 2.3.1 Exs. 9–16 Construction 7 To construct the line parallel to a given line from a point not on that line. P b A B a P A B Figure 2.23 P c A B X GIVEN: and point P not on , as in Figure 2.23a CONSTRUCT: The line through point P parallel to CONSTRUCTION: Figure 2.23b: Draw a line to become a transversal through point P and some point on . For convenience, we choose point A and draw as in Figure 2.23c. Using P as the vertex, construct the angle that corresponds to so that this angle is congruent to . It may be necessary to extend upward to accomplish this. is the desired line parallel to . Í AB Í PX Í AP ∠PAB ∠PAB Í AP Í AB Í AB Í AB Í AB In Exercises 19 to 22, complete the proof.

19. Given:

Prove:

20. Given:

bisects bisects Prove: 7 n ∠BCD CE ∠ABC BE m ∠2 + m∠3 = 90° AD 7 BC BC ⬜ DC AD ⬜ DC Exercises 2.3 In Exercises 1 to 6, 艎 and m are cut by transversal v. On the basis of the information given, determine whether 艎 must be parallel to m. 1. and 2. and 3. and 4. and 5. and 6. and In Exercises 7 to 16, name the lines if any that must be parallel under the given conditions. m ∠7 = 71.4° m ∠6 = 71.4° m ∠5 = 67.5° m ∠3 = 113.5° m ∠4 = 106° m ∠1 = 106° m ∠7 = 76° m ∠1 = 106° m ∠7 = 65° m ∠2 = 65° m ∠5 = 107° m ∠1 = 107°

18. Given:

Prove: 7 n ∠3 ⬵ ∠4 7 m m 1 2 4 3 5 6 8 7 v q 1 2 8 7 13 14 20 19 3 4 10 9 15 16 22 21 5 6 12 11 17 18 24 23 p m n Exercises 7–16

7. 8.

9. 10.

11. and

12. and

13. and

14. and are

supplementary.

15. , ,

and

16. The bisectors of and

are parallel. In Exercises 17 and 18, complete each proof by filling in the missing statements and reasons.

17. Given: and are

complementary and are complementary Prove: BC 7 DE ∠1 ∠3 ∠2 ∠1 ∠21 ∠9 m ∠18 = 70° p 7 q m ∠8 = 110° ∠9 ∠8 m ⬜ q ⬜ p m 7 n 7 m n ⬜ p ⬜ p ∠7 ⬵ ∠11 ∠9 ⬵ ∠14 ∠3 ⬵ ∠10 ∠1 ⬵ ∠20 2 1 E D 3 C B A PROOF Statements Reasons 1. 1. ? 2. 2. ? 3. 3. If two lines intersect, the vertical formed are 4. ? 4. Given 5. 5. Transitive Prop. of 6. ? 6. ? ⬵ ∠1 ⬵ ∠4 ⬵ ∠s ∠2 ⬵ ∠3 ∠1 ⬵ ∠2 7 m PROOF Statements Reasons 1. 1 and 2 are comp.; 1. ? 3 and 1 are comp. 2. 2. ? 3. ? 3. If two lines are cut by a transversal so that corr. are , the lines are 储 ⬵ ∠s ∠2 ⬵ ∠3 ∠s ∠s n 1 2 4 3 m t D C A B t 1 2 4 3 n B A E C D

21. Given: bisects

Prove:

22. Given:

Prove: MN 7 XY ∠1 ⬵ ∠2 XY 7 WZ ED 7 AB ∠3 ⬵ ∠1 ∠CDA DE

33. If two lines are parallel to the same line, then these lines

are parallel to each other. Assume three coplanar lines.

34. Explain why the statement in Exercise 33 remains true

even if the three lines are not coplanar.

35. Given that point P does not lie on line 艎, construct the line

through point P that is parallel to line 艎. 1 2 3 E C A B D 1 2 X W Y N Z M P Q A B

36. Given that point Q does not lie on , construct the line

through point Q that is parallel to . AB AB

37. A carpenter drops a plumb line from point A to .

Assuming that is horizontal, the point D at which the plumb line intersects will determine the vertical line segment . Use a construction to locate point D. AD BC BC BC Triangles Vertices Sides of a Triangle Interior and Exterior of a Triangle Scalene Triangle Isosceles Triangle Equilateral Triangle Acute Triangle Obtuse Triangle Right Triangle Equiangular Triangle Auxiliary Line Determined Underdetermined Overdetermined Corollary Exterior Angle of a Triangle The Angles of a Triangle 2.4 KEY CONCEPTS In geometry, the word union means that figures are joined or combined. In Exercises 23 to 30, determine the value of x so that line will be parallel to line m.

23. 24.

25. 26. m

∠1 = x 2 + 35 m ∠5 = x m ∠3 = x 2 m ∠7 = 5x - 3 m ∠2 = 4x + 3 m ∠5 = 4x + 5 m ∠4 = 5x m t 1 2

3 4

5 6 7 8 Exercises 28–30 D ? B C A

27. 28.

29. 30.

In Exercises 31 to 33, give a formal proof for each theorem.

31. If two lines are cut by a transversal so that the alternate

exterior angles are congruent, then these lines are parallel.

32. 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. m ∠8 = 185 - x 2 x + 1 m ∠2 = x 2 - 1x + 1 m ∠5 = 16x + 3 - x 2 - 2 m ∠3 = x + 1x + 4 m ∠5 = 2xx - 1 - 2 m ∠4 = 2x 2 - 3x + 6 m ∠2 = xx - 1 m ∠6 = x 2 - 9 m ∠5 =

3x 4

The triangle in Figure 2.24 is known as , or etc. order of letters A, B, and C being unimportant. Each point A, B, and C is a vertex of the triangle; collectively, these three points are the vertices of the triangle. , , and are the sides of the triangle. Point D is in the interior of the triangle; point E is on the triangle; and point F is in the exterior of the triangle. Triangles may be categorized by the lengths of their sides. Table 2.1 presents each type of triangle, the relationship among its sides, and a drawing in which congruent sides are marked. AC BC AB 䉭BCA, 䉭ABC A triangle symbol 䉭 is the union of three line segments that are determined by three noncollinear points. DEFINITION TABLE 2.1 Triangles Classified by Congruent Sides Type Number of Congruent Sides Scalene None Isosceles Two Equilateral Three C A B F E D Figure 2.24 TABLE 2.2 Triangles Classified by Angles Type Angles Type Angles Acute All angles acute Right One right angle Obtuse One obtuse angle Equiangular All angles congruent Triangles may also be classified according to their angles See Table 2.2. EXAMPLE 1 In not shown, , , and . Describe completely the type of triangle represented. Solution is a right isosceles triangle, or is an isosceles right triangle. 쮿 䉭HJK 䉭HJK m ∠J = 90° JK = 4 HJ = 4 䉭HJK Exs. 1–7