If Matt cleans his room, then he will go to the movie. 2. Matt does not get to go to the movie.

Exs. 5–7 To prove the statement or to complete the proof problem of the form Given: P Prove: Q by the indirect method, use the following steps:

1. Suppose that is true.

2. Reason from the supposition until you reach a contradiction. 3. Note that the supposition claiming that is true must be false and that Q must therefore be true. Step 3 completes the proof. ~Q ~Q P : Q STRATEGY FOR PROOF 왘 Method of Indirect Proof General Rule: The first statement of an indirect proof is generally “SupposeAssume the opposite of the Prove statement.” Illustration: See Example 3, which begins “Assume that .” 7 m STRATEGY FOR PROOF 왘 The First Line of an Indirect Proof The contradiction that is discovered in an indirect proof often has the form . Thus, the assumed statement has forced the conclusion , asserting that is true. Then the desired theorem the contrapositive of is also true. ~Q : ~P P : Q ~Q : ~P ~P ~Q ~P EXAMPLE 3 Complete a formal proof of the following theorem: If two lines are cut by a transversal so that corresponding angles are not congruent, then the two lines are not parallel. GIVEN: In Figure 2.13, 艎 and m are cut by transversal t PROVE: PROOF: Assume that . When these lines are cut by transversal t, the corresponding angles including and are congruent. But by hypothesis. Thus, the assumed statement, which claims that , must be false. It follows that . 7 m 7 m ∠1 ⬵ ∠5 ∠5 ∠1 7 m 7 m ∠1 ⬵ ∠5 The versatility of the indirect proof is shown in the final examples of this section. The indirect proofs preceding Example 4 contain a negation in the conclusion Prove; the proofs in the final illustrations use the indirect method to arrive at a positive conclusion. Exs. 8, 9 EXAMPLE 4 GIVEN: In Figure 2.14, plane T intersects parallel planes P and Q in lines 艎 and m, respectively PROVE: PROOF: Assume that 艎 is not parallel to m. Then 艎 and m intersect at some point A. But if so, point A must be on both planes P and Q, which means that planes P and Q intersect; but P and Q are parallel by hypothesis. Therefore, the assumption that 艎 and m are not parallel must be false, and it follows that . 쮿 7 m 7 m m 1 2 4 3 5 6 8 7 t Figure 2.13 T m P Q Figure 2.14 쮿 EXAMPLE 5 Prove the statement “The angle bisector of an angle is unique.” GIVEN: In Figure 2.15a, bisects PROVE: is the only angle bisector for PROOF: bisects , so . Suppose that [as shown in Figure 2.15b] is also a bisector of and that . m ∠ABE = 1 2 m ∠ABC ∠ABC BE m ∠ABD = 1 2 m ∠ABC ∠ABC BD ∠ABC BD ∠ABC BD B C D A a Figure 2.15 b B C D A E By the Angle-Addition Postulate, . By substitution, ; but then by subtraction. An angle with a measure of 0 contradicts the Protractor Postulate, which states that the measure of an angle is a unique positive number. Therefore, the assumed statement must be false, and it follows that the angle bisector of an angle is unique. 쮿 m ∠EBD = 0 1 2 m ∠ABC = 1 2 m ∠ABC + m∠EBD m ∠ABD = m∠ABE + m∠EBD Ex. 10 Exercises 2.2 In Exercises 1 to 4, write the converse, the inverse, and the contrapositive of each statement. When possible, classify the statement as true or false.

1. If Juan wins the state lottery, then he will be rich. 2. If ,

then .

3. Two angles are complementary if the sum of their

measures is 90°.

4. In a plane, if two lines are not perpendicular to the same

line, then these lines are not parallel. In Exercises 5 to 8, draw a conclusion where possible.

5. 1. If two triangles are congruent, then the triangles are

similar. 2. Triangles ABC and DEF are not congruent. C. ⬖ ?

6. 1. If two triangles are congruent, then the triangles are

similar. 2. Triangles ABC and DEF are not similar. C. ⬖ ? x Z x 7 2

7. 1. If , then .

2. C. ⬖ ?

8. 1. If , then .

2. C. ⬖ ?

9. Which of the following statements would you prove by

the indirect method? a In triangle ABC, if , then . b If alternate exterior alternate exterior , then 艎 is not parallel to m. c If , then or . d If two sides of a triangle are congruent, then the two angles opposite these sides are also congruent. e The perpendicular bisector of a line segment is unique.

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

indirect method, give the first statement in each proof. x = 3 x = - 2 x + 2 x - 3 = 0 ∠8 ∠1 ⬵ AC Z BC m ∠A 7 m∠B x Z 5 x = 5 x 7 3 x 7 3 x = 5 x 7 3 Indirect proofs are also used to establish uniqueness theorems, as Example 5 illustrates.

21. Given:

Prove: is not to Í EG ⬜ FH m ∠3 7 m∠4 For Exercises 11 to 14, the given statement is true. Write an equivalent but more compact statement that must be true.

11. If and are not

congruent, then and are not vertical angles.

12. If lines 艎 and m are not perpendicular, then the angles

formed by 艎 and m are not right angles.

13. If all sides of a triangle are not congruent, then the

triangle is not an equilateral triangle.

14. If no two sides of a quadrilateral figure with four sides

are parallel, then the quadrilateral is not a trapezoid. In Exercises 15 and 16, state a conclusion for the argument. Statements 1 and 2 are true.

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

triangles are not congruent. 2. Triangle ABC is congruent to triangle DEF. C. ?

16. 1. If two triangles do not have the same shape, then the

triangles are not similar. 2. Triangle RST is similar to triangle XYZ. C. ?

17. A periscope uses an indirect method of observation. This

instrument allows one to see what would otherwise be obstructed. Mirrors are located see and in the drawing so that an image is reflected twice. How are and related to each other? CD AB CD AB ‹ ‹ ∠B ∠A ∠B ∠A

20. Given:

Prove: does not bisect ∠ABC BD ∠ABD ⬵ ∠DBC D C B A

18. Some stores use an indirect method of observation. The

purpose may be for safety to avoid collisions or to foil the attempts of would-be shoplifters. In this situation, a mirror see in the drawing is placed at the intersection of two aisles as shown. An observer at point P can then see any movement along the indicated aisle. In the sketch, what is the measure of ? ∠GEF EF P E Aisle F G In Exercises 19 to 30, give the indirect proof for each problem or statement.

19. Given:

Prove: r 7 s ∠1 ⬵ ∠5 2 4 6 8 1 3 5 7 s t A B C D

22. Given:

Prove: B is not the midpoint of

23. If two angles are not congruent, then these angles are not

vertical angles.

24. If , then

.

25. If alternate interior angles are not congruent when two

lines are cut by a transversal, then the lines are not parallel.

26. If a and b are positive numbers, then .

27. The midpoint of a line segment is unique. 28. There is exactly one line perpendicular to a given line at a point on the line.

29. In a plane, if two lines are parallel to a third line, then the

two lines are parallel to each other.

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

that the corresponding angles are congruent, then the lines are parallel. 1a 2 + b 2 Z a + b x Z 5 x 2 Z 25 AD AM = CD MB 7 BC

3 4

G F H E A M C D B