Suppose that r is parallel to s

24. Given: ,

, and Find: HINT: There is a line through C parallel to both and .

25. Given: and

Find: See “Hint” in Exercise 24.

26. Given: ,

See figure on page 79. Prove:

27. In triangle ABC, line t is

drawn through vertex A in such a way that . a Which pairs of are ? b What is the sum of , , and ? c What is the sum of measures of the of ? In Exercises 28 to 30, write a formal proof of each theorem.

28. If two parallel lines are cut by a transversal, then the

alternate exterior angles are congruent.

29. If two parallel lines are cut by a transversal, then the

exterior angles on the same side of the transversal are supplementary.

30. If a transversal is perpendicular to one of two parallel

lines, then it is also perpendicular to the other line. 䉭ABC ∠s m ∠5 m ∠4 m ∠1 ⬵ ∠s t 7 BC s ⬜ t r ⬜ t r 7 s m ∠ACD m ∠BAC + m∠CDE = 93° Í AB 7 Í DE Í DE Í AB m ∠ACD m ∠EDC = 54° m ∠BAC = 42° Í AB 7 Í DE Let represent the statement “If P, then Q.” The following statements are related to this conditional statement. NOTE: Recall that represents the negation of P. ~P P : Q

31. Suppose that two lines are cut by a transversal in such a

way that corresponding angles are not congruent. Can those two lines be parallel?

32. Given: Line 艎 and point P

not on 艎 Construct:

33. Given: Triangle ABC with

three acute angles Construct:

34. Given: Triangle MNQ with

obtuse Construct:

35. Given: Triangle MNQ with

obtuse Construct: HINT: Extend .

36. Given: A line m and a

point T not on m NQ MR ⬜ NQ ∠MNQ NE ⬜ MQ ∠MNQ BD ⬜ AC Í PQ ⬜ A D B E C Exercises 24, 25 1 t 2 3

5 4

A B C P B C A M Q N Exercises 34, 35 Conditional Converse Inverse Contrapositive Law of Negative Inference Indirect Proof Indirect Proof 2.2 KEY CONCEPTS m T Suppose that you do the following: i Construct a perpendicular line r from T to line m. ii Construct a line s perpendicular to line r at point T. What is the relationship between lines s and m? Conditional or Implication If P, then Q. Converse of Conditional If Q, then P. Inverse of Conditional If not P, then not Q. Contrapositive of Conditional If not Q, then not P. ~Q : ~P ~P : ~Q Q : P P : Q