Unit 2 Angles and Proofs. Conditional Statement: Statements in if-then form are called.

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1 Unit 2 ngles and Proofs onditional Statement: in if-then form are called. The p portion after the if is The q portion after the then is xample 1: Write each statement in conditional form. a) n angle of 40 degrees is acute b) student on the high honor roll has at least a 90 average. xample 2: Identify the hypothesis and the conclusion of the following statement. I will go bowling if it rains on Tuesday. Truth Value: Statement can have a truth value of True (T) or False (F) ounterexample: counterexample to a statement is a particular example or instance of the statement that makes the statement. xample 3: Show that this conditional statement is false by finding a counter example: a) If it is February, then there are only 28 days in the month. b) If x 2 > 0, then x > 0 onverse (of a conditional): Statement formed by switching the and the xample 4: Write the converse of the statements. Is the converse true or false? a) If I study, then I get good grades. b) If two angles are complementary, they the sum of their measures is 90º. iconditional: statement that is the combination of a and its. biconditional contains the words (iff) and ***can only be written if How to write a biconditional: xample 5: Write a biconditional for the following statement, If it is Sunday, then I am watching football. 1

2 Inverse (of a conditional): statement formed by negating both the hypothesis and the conclusion. xample 6: Write the inverse of the statement: a) If it rains, then we will get wet. b) If it does not rain then we will play ultimate frisbee. ontrapositive (of a conditional) statement formed by switching N negating the hypothesis and the conclusion. **Note: a conditional and its contrapositive are logically equivalent;** ***they have the same truth value*** xample 7: Write the contrapositive of the following statement: If you live in harleston, then you live in South arolina. xample 8: Write the converse, inverse, and contrapositive of each statement. a) If you like volleyball, then you like to be at the beach. onverse: Inverse: ontrapositive: 2.5 Using Proofs in lgebra I. Name the property that justifies each statement. 1. If m = m, then m = m. 2. If x + 3 = 17, then x = xy = xy 4. If 7x = 42, then x = 6 5. If XY YZ = XM, then XY = XM + YZ 6. 2( x + 4) = 2x If m + m = 90, and m = 30, 8. If x = y + 3 and y + 3 = 10, then x = 10. then 30 + m = 90. II. omplete the reasons in each algebraic proof. 1. Prove that if 2(x 3) = 8, then x = 7. a) 2 (x 3) = 8 a) b) 2x 6 = 8 b) c) 2x = 14 c) d) x = 7 d) 2

3 2. Prove that if 3x 4 = a) 3x 4 = 1 x + 6, then x = x + 6 a) 2 b) 2(3x 4) = 2( 2 1 x + 6) b) c) 6x 8 = x + 12 c) d) 5x 8 = 12 d) e) 5x = 20 e) f) x = 4 f) 3. You can use the ngle ddition Postulate Given: m< O = 139 Solve for x and justify each step. Statement Property 1) m<o + m< O = m <O 1) 2) x + (2x + 10) = 139 2) 3) 3x + 10 = 139 3) 4) 3x = 129 4) 5) x = 23 5) 4. You can use the definition of an angle bisector. Given: LM bisects <KLN Solve for x and justify each step. Statement Property 1) LM bisects <KLN 1) 2) <KLM <MLN 2) 3) m <KLM = m <MLN 3) 4) 2x + 40 = 4x 4) 5) 40 = 2x 5) 6) 20 = x 6) 5. You can use the Segment ddition Postulate. Given: = 21 Solve for y and justify each step. Statement 1) + = 1) Property 2) 2y + 3y 9 = 21 2) 3) 5y 9 = 21 3) 4) 5y = 30 4) 5) y = 6 5) 3

4 Name the property that justifies each statement. 1. If 3x = 120, then x = If 12 =, then = If = and =, then =. 4. If y = 75 and y = m, then m = If 5 = 3x 4, then 3x 4 = If 3 x 1, then 3x 5 = If m1 = 90 and m2 = 90, then m1 = m For XY, XY = XY. 9. If F = GH and GH = JK, then F = JK. 10. If m = 90, then m1 = 60. Name the property that justifies each statement. 11. If + = +, then =. 12. m = m 13. If XY = PQ and XY = RS, then PQ = RS. 14. If x 5, then x = If 2x = 9, then x = ach statement below has reference to one of the two diagrams given. Suppose each statement is TRU. Name the reason for each true statement. efinitions must be written out! 1. If 1 2, then. 2. m1 + m2 = mmxy 3. = 4. If =, then + = + 5. If = and = F, then = F 6. If is the midpoint of. 4

5 7. If m = m, then m = m. 8. If = and =, then = = 10. If bisects MXY, then If m1 = m2 and m2 = m3, then m1 = m F = F 13. If + = and = F, then = F =. l If is the midpoint of, then line l bisects. F Fill in the given statements or reasons 16. Given: XY = YZ Prove: Y is the midpoint of XZ 1) XY = YZ 1) 2) XY YZ 2) 3) Y is the midpoint of XZ 3) Mini Proofs: Fill in the reasons below for each proof..) Given: is the midpoint of. is the midpoint of. STTMNTS Prove: + = + RSONS 1. is the midpoint of. 2. is the midpoint of = = =

6 ) P Q R S T Given: PQ = RS Prove: PQ + QR = QS STTMNTS RSONS 1. PQ = RS QR + RS = QS QR + PQ = QS 3. ) Given: l bisects l bisects F Prove: + = + F is the midpoint of F. 2. is the midpoint of F 4. = 4. = F 5. + = + F 5. l F 17. Given: bisects Prove: = 1) bisects 1) 2) is the midpoint 2) 3) 3) 4) = 4) 18. Given: M is the midpoint of PQ Prove: PM MQ Q M P 1) M is the midpoint of PQ 1) 2) PM MQ 2) 3) PM MQ 3) 6

7 X 19. Given: XP YP Prove: m bisects XY 1) XP YP 1) m P Y 2) P is the midpoint of XY 2) 3) m bisects XY 3) 5. Given: is midpoint of is midpoint of Prove: + 1) is midpoint of 1) is midpoint of 2) 2) 3) 3) 4) + 4) Practice with Segment Proofs 1.Given: RT SU Prove: RS = TU 1. RT SU RT = SU RT = RS + ST 3. SU = ST + TU 4. RS + ST = ST + TU ST = ST RS = TU Given: = F, = F Prove: = R S T U F 1. = F 1. = F 2. + = F + F = 3. = F + F 4. = 4. Given: X = Y X = Y X Y 7

8 Prove: 1. X = Y 1. X = Y 2. X + X = Y + Y X + X = 3. Y + Y = 4. = Fill in the blanks to complete a two-column proof of one case of the ongruent Supplements Theorem. Given: 1 and 2 are supplementary, and 2 and 3 are supplementary. Prove: 1 3 Proof: Use the given plan to write a two-column proof. Given: 1 and 2 are supplementary, and 1 3 Prove: 3 and 2 are supplementary. 8

9 Use the given plan to write a two-column proof if one case Theorem. Given: 1 and 2 are complementary, and 2 and 3 are complementary. Prove: 1 3 of ongruent omplements 2.Given: HKJ is a straight angle. KI bisects HKJ. Prove: IKJ is a right angle. Proof: 1. a. 1. Given 2. mhkj b. 3. c. 3. Given 4. IKJ IKH mikj mikh d. 6. dd. Post. 7. 2mIKJ mikj IKJ is a right angle. 9. f. 3. Given: Q is the midpoint of PR. T is the midpoint of SV. PQ = ST Prove: QR TV 4. Given: = F = P Q R S T V Prove : F F 9

10 5. Given: RS TV Prove: RT = SV R S T V 6. Given: PR = SV PQ = TV Prove: QR = ST P Q R S T V 7. Given: I bisects NH Prove: N H I 9. Given: 1 is a supplement of 2 Prove: 1 3 m t r 10. Given: Prove: 11. Given: 1 4 Prove: Given: MH HM RMH RHM Prove: MR HR M H 13. Given: F bisects Prove: F R 10 F

11 14. Given: ; 1 is a complement of 3 Prove: Given: O ; 2 3 Prove: O 16. Given: Prove: and are complementary 17. Given: XY = YZ Prove: Y is the midpoint of XZ X 18. Given: Line m bisects XY Prove: XP = YP P m 19. Given: is the midpoint of is the midpoint of Prove: + = + Y 20. Given: PQ = MN MN = QR Prove: Q is the midpoint of PR 21. Given: line k bisects line k bisects F Prove: + = + F 22. Given: is the midpoint of Prove: + = P Q R M N F k 11 F