Given: 2x Given: TEST 534

30. Given:

⬔ABC and See figure for Exercise 29. Prove: m ⬔ABD = m ⬔ABC - m⬔DBC PROOF Statements Reasons 1. ⬔ABC and 1. ? 2. m ⬔ABD + m ⬔DBC 2. ? = m ⬔ABC 3. m ⬔ABD = m ⬔ABC 3. ? - m⬔DBC In Exercises 31 and 32, fill in the missing statements and reasons.

31. Given: M-N-P-Q

on Prove: MN + NP + PQ = MQ PROOF Statements Reasons 1. ? 1. ? 2. MN + NQ = MQ 2. ? 3. NP + PQ = NQ 3. ? 4. ? 4. Substitution Property of Equality

32. Given:

⬔TSW with and Prove: m ⬔TSW = m ⬔TSU + m ⬔USV + m ⬔VSW SV SU MQ BD BD In Exercises 27 to 30, fill in the missing reasons for each geometric proof.

27. Given: D

-E-F on Prove: DE = DF - EF PROOF Statements Reasons 1. D-E-F on 1. ? 2. DE + EF = DF 2. ? 3. DE = DF - EF 3. ?

28. Given:

E is the midpoint of Prove: DE = DF PROOF Statements Reasons 1. E is the midpoint of 1. ? 2. DE = EF 2. ? 3. DE + EF = DF 3. ? 4. DE + DE = DF 4. ? 5. 2DE = DF 5. ? 6. DE = DF 6. ?

29. Given: bisects

⬔ABC Prove: m ⬔ABD = m ⬔ABC 1 2 BD 1 2 DF 1 2 DF Í DF Í DF D F E Exercises 27, 28 PROOF Statements Reasons 1. ? 1. ? 2. m ⬔TSW = m ⬔TSU 2. ? + m ⬔USW 3. m ⬔USW = m ⬔USV 3. ? + m ⬔VSW 4. ? 4. Substitution Property of Equality B A D C Exercises 29, 30 PROOF Statements Reasons 1. bisects ⬔ABC 1. ? 2. m ⬔ABD = m ⬔DBC 2. ? 3. m ⬔ABD + m ⬔DBC 3. ? = m ⬔ABC 4. m ⬔ABD + m ⬔ABD 4. ? = m ⬔ABC 5. 2m ⬔ABD = m ⬔ABC 5. ? 6. m ⬔ABD = m ⬔ABC 6. ? 1 2 BD T U V W S M P N Q

33. When the Distributive Property is written in its symmetric

form, it reads a b + a c = a b + c . Use this form to rewrite 5x +

5y.

34. Another form of the Distributive Property see Exercise

33 reads b a + c a = b + c a. Use this form to rewrite 5x + 7x. Then simplify.

35. The Multiplication Property of Inequality requires that we

reverse the inequality symbol when multiplying by a negative number. Given that -7 5, form the inequality that results when we multiply each side by -2.

36. The Division Property of Inequality requires that we

reverse the inequality symbol when dividing by a negative number. Given that 12 -4, form the inequality that results when we divide each side by -4.

37. Provide reasons for this proof. “If a

= b and c = d , then a + c = b + d .” PROOF Statements Reasons

1. a

= b 1. ?

2. a

+ c = b + c 2. ? 3. c = d 3. ? 4. a + c = b + d 4. ?

38. Write a proof for: “If a

= b and c = d , then a - c = b - d .” Informally, a vertical line is one that extends up and down, like a flagpole. On the other hand, a line that extends left to right is horizontal. In Figure 1.59, is vertical and j is horizontal. Where lines and j intersect, they appear to form angles of equal measure. Vertical Lines Horizontal Lines Perpendicular Lines Relations: Reflexive, Symmetric, and Transitive Properties Equivalence Relation Perpendicular Bisector of a Line Segment Relationships: Perpendicular Lines 1.6 KEY CONCEPTS Perpendicular lines are two lines that meet to form congruent adjacent angles. DEFINITION Perpendicular lines do not have to be vertical and horizontal. In Figure 1.60, the slanted lines m and p are perpendicular m › p. As we have seen, a small square is of- ten placed in the opening of an angle formed by perpendicular lines to signify that the lines are perpendicular. Example 1 provides a formal proof of the relationship between perpendicular lines and right angles. Study this proof, noting the order of the statements and reasons. The numbers in parentheses to the left of the statements refer to the earlier statements upon which the new statement is based. If two lines are perpendicular, then they meet to form right angles. THEOREM 1.6.1 j Figure 1.59 m p Figure 1.60 General Rule: Make a drawing that accurately characterizes the “Given” information. Illustration: For the proof of Example 1, see Figure 1.61. STRATEGY FOR PROOF 왘 The Drawing for the Proof