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1) Given: 1 and 4 are supplementary.Prove: a b 1 and 4 aresupplementary 2 and 3 are supplementaryGIVENSubstitution Property 1 2 and 3 4a ll bCONVERSE SSIA THMVAT2) Given: q r, r s, b q, and a sProve: a bProof: Because it is given that q r and r s, then q s by the TRANSITIVE PROPERTY OFPARALLEL LINES .ANGLES THM .This means that 1 2 because the CORRESPONDINGBecause b q, m 1 90. So, m 2 90 . This means s b, by definition ofperpendicular lines. It is given that a s, so a b BECAUSE IF TWO LINES ARE PERPENDICULARTO THE SAME LINES, THOSE LINES MUST BE PARALLEL .

3) GIVEN: g h, 1 2PROVE: p r4)Statements1) g hReasons1. GIVEN2) 1 32. CORRESPONDING ANGLES THEOREM (CAT)3) 1 23. GIVEN4) 2 34. TRANSITIVE PROPERTY5) p r5. CONVERSE AEA THEOREMGiven:m, a bProve: 1 5StatementsReasons1. Given1.m, a b2. 1 22. VERTICAL ANGLES THEOREM (VAT)3. 2 and 3 are supplementary.3. SAME SIDE INTERIOR ANGLES THM (SSIA THM)4. 3 and 4 are supplementary.4. SAME SIDE INTERIOR ANGLES THM (SSIA THM)5. 2 45. CONGRUENT SUPPLEMENTS THEOREM (IF TWOANGLES ARE SUPPLEMENTARY TO THE SAME ANGLE THOSEANGLES ARE CONGRUENT)6. 1 47. 4 57. VERTICAL ANGLES THEOREM (VAT)8. 1 58. TRANSITIVE PROPERTY6. TRANSITIVE PROPERTY

5) Given: 1 and 2 are supplementary; x yProve: q rx ll yGIVEN 2 and 3 aresupplementarySSIA THEOREM 1 3 SUPPLEMENTS THM 1 and 2 aresupplementaryq ll r.CONVERSE AEA THMGIVEN6)Given: 1 4Prove: 2 3Proof: 1 4 because it is given. 1 2 by the VERTICAL ANGLES THEOREM(VAT) . 2 4 by the TRANSITIVE PROPERTY . 3 4 by theVAT . It follows that 2 3 by the TRANSITIVE PROPERTY .

7) GIVEN: p q, q rPROVE: p rStatements1. p qReasons1) GIVEN2. 1 is a right angle.2) DEFINITION OF PERPENDICULAR3. m 1 90 3) DEFINITION OF RIGHT ANGLE4. q r4) GIVEN5. 1 25) CORRESPONDING ANGLES THEOREM (CAT)6. m 1 m 26) DEFINITION OF CONGRUENT7. m 2 90 7)SUBSTITUTION8. 2 is a right angle.8)DEFINITION OF RIGHT ANGLE9. p r9)DEFINITION OF PERPENDICULAR8) GIVEN: g h, 1 2PROVE: p rStatements1. g hReasons1. GIVEN2. 1 32. CORRESPONDING ANGLES THOREM (CAT)3. 1 23. GIVEN4. 2 34. TRANSITIVE PROPERTY5. p r5. CONVERSE CORRESPONDING ANGLES THEOREM

9) Given: 1 is supplementary to 2Prove:1lmm23 1 and 2 aresupplementaryGIVEN 1 3l ll mCongruent SupplementsTheorem 2 and 3 aresupplementaryCONVERSE AEA THMLINEAR PAIR10) Write a paragraph proof.Given: PQS and QSR are supplementary.Prove:PROOF: IT IS GIVEN THAT PQS AND QSR ARE SUPPLEMENTARY. THUS BYCONVERSE SSIA, ⃡⃡. IT IS ALSO GIVEN THAT ⃡⃡ ONP AND QPN ARE SUPPLEMENTARY. THEREFORE ⃡TRANSITIVE PROPERTY OF PARALLEL LINES, ⃡⃡AND ⃡⃡.⃡. BY THETHUS

11) GIVEN: n m, 1 2PROVE: p rStatements1) n mReasons1. GIVEN2) 1 32. ALTERNATE INTERIOR ANGLES THEOREM3) 1 23. GIVEN4) 2 34. TRANSITIVE PROPERTY5) p r5. CONVERSE AIA THEOREM12) Given: 1 2Prove: 3 4Statements1) 1 2Reasons1) Given2) m 1 m 3 m 5 1802) DEFINITION OF STRAIGHT ANGLE3) m 1 m 3 90 1803) SUBSTITUTION PROPERTY4) m 1 m 3 904) SUBTRACTION PROPERTY5) m 4 m 2 m 55) VERTICAL ANGLES THOREM6) m 4 m 2 906) SUBSTITUTION PROPERTY7) m 4 m 1 907) SUBSTITUTION PROPERTY (SINCE 1 2 )8) m 1 m 3 m 4 m 18) TRANSITIVE PROPERTY9) m 4 m 39) SUBTRACTION PROPERTY10) 3 410) DEFINITION OF CONGRUENT

13) Write a paragraph proof.Given: a b , a , b mProve:PROOF:ma ll b and al means that lb since a line perpendicular toparallel lines is perpendicular to both lines (thm 3-9). Since l b and we aregiven b m, then l ll m since two lines perpendicular to the same line mustbe parallel to each other (thm 3-8)14) Complete the two-column proof.GIVEN: q rPROVE: 1 3Statements1. q rReasons1.GIVEN2. 1 22.VERTICAL ANGLES THEOREM3. 2 33.CORRESPONDING ANGLES THEOREM4. 1 34.TRANSITIVE PROPERTY

15) GIVEN: g h, m 1 122 , m 4 122 1 3PROVE: p rStatements1. g h2. m 1 122 , m 4 122 3. m 1 m 4Reasons1) GIVEN2) GIVEN3) TRANSITIVE PROPERTY4. 1 44) DEFINITION OF CONGRUENT5. 1 35) GIVEN6. 3 46) TRANSITIVE PROPERTY7. p r7) CONVERSE ALTERNATE INTERIOR ANGLES THM16) GIVEN: q r, p tPROVE: 1 3StatementsReasons1. p t1) GIVEN2. l 22) ALERNATE EXTERIOR ANGLES THEOREM3. q r3) GIVEN4. 2 34) CORRESPONDING ANGLES THEOREM5. 1 35) TRANSITIVE PROPERTY

17) Write a flow proofGiven: 2 and 3 are supplementary.Prove: c ll d 2 & 3 ARE SUPPLEMENTARYGIVEN 1 & 2 ARE SUPPLEMENTARYc ll d 1 3( SUPPLEMENTSTHM)(CONVERSE AEATHM)(LINEAR PAIR)18)VERTICAL ANGLES THEOREMGIVENSAME SIDE INTERIOR ANGLES THEOREMGIVENALTERNATE INTERIOR ANGLES THEOREMSUBSTITUTION PROPERTY

19) Write a paragraph proof of Theorem 3-9:PROOF:WE ARE GIVEN THATTHUS ANGLES 1 AND 2 ARE RIGHT ANGLESAND ALL RIGHT ANGLES ARE CONGRUENT. SINCE ANGLES 1 AND 2 ARE CORRESPONDINGANGLES, LINE N MUST BE PARALLEL TO LINE O BY THE CONVERSE CORRESPONDING ANGLESTHEOREM.20) GIVEN: 1 3, 1 and 2 are supplementaryPROVE: p rStatements1. g hReasons1. GIVEN2. 1 and 2 are supplementary2. GIVEN3. 1 33. GIVEN4. 3 and 2 are supplementary4. SUBSTITUTION5. p r5. CONVERSE SSIA THM

21) 2 3(GIVEN)a ll b(CONVERSE AEATHM)22) Complete the paragraph proof of Theorem 3-8Given: d ll e, e ll fProve: d ll fProof: Because it is given that d ll e, then 1 is supplementary to 2 by the SAME SIDEINTERIOR ANGLES THEOREM . Because it is given that e ll f , then 2 3 by theCORRESPONDING ANGLES THEOEM. Thus, by substitution 1 is supplementary to 3 .And by CONVERSE CORRESPONDING ANGLES THEOREM d ll f.

23)VERTICAL ANGLES THEOREMGIVENCORRESPONDING ANGLES THEOREMSAME SIDE INTERIOR ANGLES THEOREMSUBSTITUTION PROPERTY24)GIVEN: 1 2, 3 4PROVE: n pSTATEMENTSREASONS1. 1 22. l m3. 4 53. 3 44. 3 54. n p1) GIVEN2) CONVERSE CORRESPONDING ANGLES THEOREM3) AIA THEOREM3) GIVEN4) TRANSITIVE PROPERTY4) CONVERSE CORRESPONDING ANGLES THEOREM

25) Write a flow proofl ll n(GIVEN) 8 4 12 4 12 HEOREM)j ll k(CONVERSE CAT)(GIVEN)26)PROOF:SINCE WE ARE GIVEN THAT a ll c and b ll c, then a ll b by the TRANSITIVEPROPERTY OF PARALLEL LINES. THUS BY THE ALTERNATE INTERIOR ANGLESTHEOREM 1 2. SINCE WE ARE GIVEN m 2 65, then m 1 65 BY THEDEFINITION OF CONGRUENT.

27) Given l 2Prove QPS and l are right angles28)Statements1. l 2Reasons1. GIVEN2. PS PQ2. IF SUPPLEMENTARY ANGLES ARECONGRUENT, THEN THE LINES AREPERPENDICULAR3. QPS and 1 are right angles.3.DEFINITION OF RIGHT ANGLES.GIVEN: j k, 1 2PROVE: r sStatements1. j kReasons1.GIVEN2. 1 52.CAT (if lines are parallel, then Corrsp are congru)3. 1 23.GIVEN4. 5 23.TRANSITIVE PROPERTY5.CONVERSE AIA THM (if alt interior angles are5. r scongru, then lines are parall)

29) Complete the paragraph proof of the Perpendicular Transversal Theorem (Thm 3-10)Proof: Since y ll z, m 1 90 by the CORRESPONDING ANGLES THEOREM .By definition of PERPENDICULAR lines, xz .30) GIVEN: CA ED ,m FED m GCA 45 PROVE: EF CGStatements1. CA ED2. CBE FED3. m FED m GCA 45 Reasons1.GIVEN2.AIA THM (if lines are parallel, then AIA are congru)3.GIVEN3. FED GCA3.DEFINITION OF CONGRUENT4. CBE GCA3.TRANSITIVE PROPERTY5.CONVERSE AIA THM (if alt interior angles are5. EF CGcongru, then lines are parall)

31) Given l 2Prove 3 and 4 are complementary.PROOF: WE ARE GIVEN THAT l 2. SINCE 1 AND 2 FORM A STRAIGHTANGLE, m 1 m 2 90 . WE ALSO KNOW BY THE VERTICAL ANGLE THEOREMTHAT l IS CONGRUENT TO 3 AND 4 COMBINED. THUS m l m 3 m 4. USINGSUBSTITUTION WE HAVE 90 m 3 4. THUS 3 AND 4 ARE COMPLEMENTARYBY THE DEFINITION OF COMPLEMENTARY.32) Given:m, a b, a Prove: b mStatements1.m, a b, a Reasons1.GIVEN2. 3 IS A RIGHT ANGLE2. CORRESPONDING ANGLES THEOEM3. 3 AND 4 ARESUPPLEMENTARY3.SSIA THM (if lines are parallel, then SSIA are SUPP)4. 4 IS A RIGHT ANGLE4.DEFINITION OF SUPPLEMENTARY5. b m5.DEFINITION OF PERPENDICULAR

33)PROOF: WE ARE GIVEN THATTHUS a ll c BY THE TRANSITIVE PROPERTY OFPARALLEL LINES. WE ARE ALSO GIVEN THAT. IF A LINE IS PERPENDICULARTO ONE OF TWO PARALLEL LINES, THEN THE LINE IS PERPENDICULAR TO BOTH LINES.THUS.34)Statements1. r ll sReasons1.GIVEN2. 1 62. CORRESPONDING ANGLES THEOEM2. 8 62. VERTICAL ANGLES THEOREM2. 1 82. TRANSITIVE PROPERTY

35) Write a flow proofj ll k(GIVEN)m 8 m 9 180(GIVEN) 9 3(AIA THEOREM)m 8 m 3 180(SUBSTITUTIONPROPERTY)l ll n(CONVERSE SSIATHM)36) Complete the paragraph proof of Theorem 3-8 for 3 coplanar linesProof: Since l ll k, 2 1 by the CORRESPONDING ANGLES THEOREM . Since m ll k, 3 1 for the same reason. By the Transitive property of congruence, 2 3 . Thusby the CONVERSE CORRESPONDING ANGLES THEOREM, l ll m.

37)PROOF: WE ARE GIVEN. IF TWO LINES ARE PERPENDICULAR TO THESAME LINE THEN THE LINES ARE PARALLEL, therefore a ll c. WE ARE ALSO GIVEN THATc ll d, THUS BY THE TRANSITIVE PROPERTY OF PARALLEL LINES, a ll d.38) Write a 2-column proof:Given: a b, x yProve: 4 is supplementary to 15Statements1. a bReasons1.GIVEN2. 4 122. CORRESPONDING ANGLES THEOEM3. x y3. GIVEN4. 12 164. CORRESPONDING ANGLES THEOEM5. 4 164. TRANSITIVE PROPERTY6. 16 is supplementary to 55. LINEAR PAIR7. 4 is supplementary to 55. SUBSTITUTION PROPERTY

39) Use the diagram to answer the following:a)There isn’t a “special” angle relationship directly between 1 and 2, but if we keep line C’sslope the same and move it above line A, then 1 and 2 become same side interior angles.And since we are given that 1 and 2 are supplementary, then lines A and C are parallel by theConverse SSIA theorem.b)We are given on the diagram that Line B is parallel to Line C. So if Line A is parallel to Line C,then by the transitive property of parallel lines, Line A is parallel to Line B.

19) write a paragraph proof of theorem 3-9: proof: we are given that thus angles 1 and 2 are right angles and all right angles are congruent. since angles 1 and 2 are corresponding angles, line n must be parallel to line o by the converse corresponding angles theorem.