Suppose that is true.

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 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