8.1 Algebraic Proofs (G.CO.9) Properties of Equality Property Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property Notes If, then. If, then. If, then. If, then. If a(b + c), then a(b + c) =. If then a may be by b in any expression or equation. For any real number a,. (A value always will equal itself!) If, then. If and, then. Using the Properties of Equality Properties of equality can be used to justify steps in solving an equation. NEW! Two-Column Proof: A common format used to organize a proof. Left Side: List the (or steps). Right Side: List the that justify each step. What can be used as reasons?,,,, ❶ Given: 4x 1 = 27; Prove: x = 7 1. 4x 1 = 27 1. Given 2. 4x = 28 2. 3. x = 7 3.
❷ a Given: + 2 = 5 ; Prove: a = -18 6 a 1. + 2= 5 6 a 2. 3 6 = 1. Given 2. 3. a = -18 3. ❸ Given: -9(2x 3) = 63; Prove: x = -2 1. -9(2x 3) = 63 1. Given 2. -18x + 27 = 63 2. 3. -18x = 36 3. 4. x = -2 4. ❹ Given: 6x + 7 = 8x 17; Prove: x = 12 1. 6x + 7 = 8x 17 1. Given 2. 7 = 2x 17 2. 3. 24 = 2x 3. 4. 12 = x 4. 5. x = 12 5. ❺ Given: -7(x + 2) + 4x = 6(2x 4); Prove: x = 2/3 1. -7(x + 2) + 4x = 6(2x 4) 1. Given 2. -7x 14 + 4x = 12x 24 2. 3. -3x 14 = 12x 24 3. 4. -15x 14 = -24 4. 5. -15x = -10 5. 6. x = 2/3 6.
8.2 Segment Proofs (G.CO.9, 12) Segments Proofs Reference Properties of Equality Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property The properties above may only be used with EQUAL signs. The following properties of congruence can be applied to statements with congruence symbols: Properties of Congruence Reflexive Property of Congruence For any segment AB,. Symmetric Property of Congruence Transitive Property of Congruence If, then. If and, then. Definitions Congruence Segments are congruence if and only if they have the same measure: If, then. If, then. Midpoint The midpoint of a segment divides the segment into 2 equal (congruent) parts. If M is the midpoint of AB, then Postulates If A, B, and C are collinear points and B is between A and C: Segment Addition Postulate A B C then:
Practice! Justify each of the following statements using a property of equality, property of congruence, definition, or postulate. 1. If PQ = PQ, then PQ PQ 2. If K is between J and L, then JK + KL = JL 3. EF EF 4. If RS = TU, then RS + XY = TU + XY 5. If AB = DE, then DE = AB 6. If Y is the midpoint of XZ, then XY = YZ 7. If FG HI and HI JK, then FG JK 8. If AB + CD = EF + CD, then AB = EF 9. If PQ + RS = TV and RS = WX, then PQ + WX = TV 10. If LP = PN, and L, P, and N are collinear, then P is the midpoint of LN 11. If UV UV, then UV = UV 12. If CD + DE = CE, then CD = CE DE Property Bank: Properties of Equality: Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property Properties of Congruence: Reflexive Property Symmetric Property Transitive Property Definitions: Congruence Midpoint Postulates: Segment Addition Postulate
Segments Proofs Directions: Complete the proofs below by giving the missing statements and reasons. 1 Given: E is the midpoint of DF Prove: 2DE = DF 1. E is the midpoint of DF 1. 2. DE = EF 2. 3. DE + DE = DE + EF 3. 4. 2DE = DE + EF 4. 5. DE + EF = DF 5. 6. 2DE = DF 6. D E F 2 Given: KL LN, LM LN Prove: L is the midpoint of KM K L N M 1. KL LN, LM LN 1. 2. KL = LN, LM = LN 2. 3. KL = LM 3. 4. L is the midpoint of KM 4. P U 3 Given: PQ TQ, UQ QS Prove: PS TU T Q S 1. PQ TQ, UQ QS 1. 2. PQ = TQ, UQ = QS 2. 3. PQ + QS = PS; TQ + QU = TU 3. 4. TQ + QS = PS 4. 5. TQ + QS = TU 5. 6. PS = TU 6. 7. PS TU 7.
J N 4 Given: K is the midpoint of JL, M is the midpoint of LN, JK = MN Prove: KL LM K L M 1. K is the midpoint of JL, M is the midpoint of LN 1. 2. JK = KL, LM = MN 2. 3. JK = MN 3. 4. MN = KL, LM = MN 4. 5. LM = KL 5. 6. KL = LM 6. 7. KL LM 7. 5 Given: XY UV, YZ TU Prove: XZ TV 1. XY UV, YZ TU 1. 2. XY = UV, YZ = TU 2. 3. XY + YZ = XZ, TU + UV = TV 3. 4. UV + YZ = XZ, YZ + UV = TV 4. 5. XZ = TV 5. 6. XZ TV 6. X Y Z T U V 6 Given: YW YZ, XY VY Prove: XZ VW V X Y Z W 1. WY YZ, XY VY 1. 2. WY = YZ, XY = VY 2. 3. XY + YZ = XZ 3. 4. VY + YW = XZ 4. 5. VY + YW = VW 5. 6. XZ = VW 6. 7. XZ VW 7.
Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Angle Proofs Reference Properties of Equality 8.3 Angle Proofs (G.CO.9, 12) Substitution Property Reflexive Property Symmetric Property Transitive Property Definitions Properties of Congruence Reflexive Property Symmetric Property Transitive Property Congruence Angle Bisector Complementary Angles Supplementary Angles Perpendicular a Right Angle m A = m B A B An angle bisector divides an angle into two equal parts. Complementary Sum is 90. Supplementary Sum is 180. Perpendicular lines form right angles. A right angle = 90. Postulates Angle Addition Postulate B A D C m ABD + m DBC = ABC Theorems Vertical Angles Theorem Complement Theorem Supplement Theorem Congruent Complements Theorem Congruent Supplements Theorem If two angles are vertical, then they are congruent. If two angles form a right angle, then they are complementary. Right Angle Complementary If two angles form a linear pair, then they are supplementary. Linear pair Supplementary If A is complementary to B and C is complementary to B, then A C If A is supplementary to B and C is supplementary to B, then A C
Practice! Justify each of the following statements using a definition,, theorem or postulate. 1. If A is a right angle, then m A = 90 2. If X is supplementary to Y and X is supplementary to Z, then X Z. 3. If then, 1 2 1 2 4. If m P + m Q = 90, then P and Q are complementary. 5. If M and N form a right angle, then then M and N are complementary. m 6. Given: If l m, then 1 l 1 is a right angle. 7. If W and X are supplementary, then m W + m X = 180. 8. If L is complementary to M and N is complementary to M, then L N. 9. If A and B form a linear pair, then then A and B are supplementary. 10. If N and P are complementary, then m N + m P = 90. 11. Given: K J M L m JKM + m MKL = m JKL 12. If m R = m S, then R T
ANGLE PROOFS Directions: Complete the proofs below by giving the missing statements and reasons. ❶ Given: PQR is a right angle Prove: PQS and SQR are complementary 1. PQR is a right angle 1. 2. m PQR = 90 2. 3. m PQS + m SQR = m PQR 3. 4. m PQS + m SQR = 90 4. 5. PQS and SQR are complementary 5. P Q S R ❷ Given: 2 3; 1 and 2 form a linear pair Prove: 1 and 3 are supplementary 1 2 3 1. 2 3 1. 2. m 2 = m 3 2. 3. 1 and 2 form a linear pair 3. 4. 1 and 2 are supplementary 4. 5. m 1 + m 2 = 180 5. 6. m 1 + m 3 = 180 6. 7. 1 and 3 are supplementary 7. ❸ Given: 1 and 2 form a right angle; m 1 + m 3 = 90 Prove: 2 3 1. 1 and 2 form a right angle 1. 2. 1 and 2 are complementary 2. 3. m 1 + m 3 = 90 3. 4. 1 and 3 are complementary 4. 5. 2 3 5. 2 3 1
❹ Given: BE bisects ABD; BD bisects EBC Prove: ABE DBC 1. BE bisects ABD 1. 2. ABE EBD 2. 3. BD bisects EBC 3. 4. EBD DBC 4. 5. ABE DBC 5. A E D C B ❺ Given: RSU VST Prove: RSV UST U R S V T 1. RSU VST 1. 2. m RSU = m VST 2. 3. m RSU + m USV = m RSV 3. 4. m VST + m USV = m UST 4. 5. m RSU + m USV = m UST 5. 6. m RSV = m UST 6. 7. RSV UST 7. ❻ Given: 1 and 2 are complementary 3 and 4 are complementary Prove: 1 4 1 2 3 4 1. 1 and 2 are complementary 1. 2. 3 and 4 are complementary 2. 3. m 1 + m 2 = 90 3. 4. m 3 + m 4 = 90 4. 5. 2 3 5. 6. m 2 = m 3 6. 7. m 1 + m 2 = m 3 + m 4 7. 8. m 1 + m 3 = m 3 + m 4 8. 9. m 1 = m 4 9. 10. 1 4 10.
8.4 Parallel Line Proofs (G.CO.9, 12) Proving Lines Parallel You can prove lines are parallel by the following reasons: 1 2 4 3 5 6 8 7 l m Corresponding Angles Converse Alternate Interior Angles Converse Alternate Exterior Angles Converse Consecutive Interior Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Example: If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Example: If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Example: If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. Example: Practice! Given the following information, determine which lines, if any, are parallel. State the converse that justifies your answer. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a b c d 1 Given Parallel Lines Converse a. 2 4 b. 5 10 c. m 6 + m 10 = 180 d. 1 14 e. m 14 + m 15 = 180 f. 11 16 g. 4 15 h. 10 12 i. m 9 + m 13 = 180 j. 2 7 k. 6 11
1 2 5 6 p 4 3 8 7 9 10 13 14 12 11 16 15 17 18 19 20 21 22 23 24 q r s t 2 Given Parallel Lines Converse a. 9 22 b. m 8 + m 13 = 180 c. 1 17 d. 16 20 e. 5 24 f. m 4 + m 17 = 180 g. 10 13 h. 3 22 i. 5 15 j. m 11 + m 16 = 180 Proofs: Complete the proof below by filling in the missing reasons. Given: 4 and 5 are supplementary Prove: j k 3 1 2 4 3 5 6 8 7 j k Stations 1. 4 and 5 are supplementary 1. 2. m 4 + m 5 = 180 2. 3. j k 3. 4 4 Given: 1 and 2 form a linear pair; 1 and 4 are supplementary 1 Prove: a b 2 3 7 8 5 6 a b Stations 1. 1 and 2 form a linear pair 1. 2. 1 and 2 are supplementary 2. 3. 1 and 4 are supplementary 3. 4. 2 4 4. 5. a b 5.