1) Given: 1 and 4 are supplementary. Prove: a b GIVEN VAT. Proof: Because it is given that q r and r s, then q s by the TRANSITIVE PROPERTY OF

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1 1) Given: 1 and 4 are supplementary. Prove: a b 1 and 4 are supplementary GIVEN 2 and 3 are supplementary a ll b 1 2 and 3 4 Substitution Property CONVERSE SSIA THM VAT 2) Given: q r, r s, b q, and a s Prove: a b Proof: Because it is given that q r and r s, then q s by the TRANSITIVE PROPERTY OF PARALLEL LINES. This means that 1 2 because they are CORRESPONDING ANGLES. Because b q, m 1 = 90. So, m 2 =_90_. This means s b, by definition of perpendicular lines. It is given that a s, so a b BECAUSE IF TWO LINES ARE PERPENDICULAR TO THE SAME LINES, THOSE LINES MUST BE PARALLEL.

2 3) GIVEN: g h, 1 2 PROVE: p r 1) g h 1. GIVEN 2) CORRESPONDING ANGLES THEOREM (CAT) 3) GIVEN 4) TRANSITIVE PROPERTY 5) p r 5. CONVERSE AEA THEOREM 4) Given: m, a b Prove: m, a b 1. Given 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) CONGRUENT SUPPLEMENTS THEOREM (IF TWO ANGLES ARE SUPPLEMENTARY TO THE SAME ANGLE THOSE ANGLES ARE CONGRUENT) TRANSITIVE PROPERTY VERTICAL ANGLES THEOREM (VAT) TRANSITIVE PROPERTY

3 5) Given: 1 and 2 are supplementary; x y Prove: q r x ll y 2 and 3 are supplementary GIVEN SSIA THEOREM 1 3 q ll r 1 and 2 are supplementary SUPPLEMENTS THM. CONVERSE AEA THM GIVEN 6) Given: 1 4 Prove: 2 3 Proof: 1 4 because it is given. 1 2 by the VERTICAL ANGLES THEOREM (VAT). 2 4 by the TRANSITIVE PROPERTY. 3 4 by the VAT. It follows that 2 3 by the TRANSITIVE PROPERTY.

4 7) GIVEN: p q, q r PROVE: p r 1. p q 1) GIVEN 2. 1 is a right angle. 2) DEFINITION OF PERPENDICULAR 3. m 1 = 90 3) DEFINITION OF RIGHT ANGLE 4. q r 4) GIVEN ) CORRESPONDING ANGLES THEOREM (CAT) 6. m 1= m 2 6) DEFINITION OF CONGRUENT 7. m 2 = 90 7)SUBSTITUTION 8. 2 is a right angle. 8)DEFINITION OF RIGHT ANGLE 9. p r 9)DEFINITION OF PERPENDICULAR 8) GIVEN: g h, 1 2 PROVE: p r 1. g h 1. GIVEN CORRESPONDING ANGLES THOREM (CAT) GIVEN TRANSITIVE PROPERTY 5. p r 5. CONVERSE CORRESPONDING ANGLES THEOREM

5 9) Given: 1 is supplementary to 2 Prove: m 1 and 2 are supplementary l m GIVEN 1 3 l ll m 2 and 3 are supplementary LINEAR PAIR Congruent Supplements Theorem CONVERSE AEA THM 10) Write a paragraph proof. Given: PQS and QSR are supplementary. Prove: PROOF: IT IS GIVEN THAT PQS AND QSR ARE SUPPLEMENTARY. THUS BY CONVERSE SSIA,. IT IS ALSO GIVEN THAT AND THUS ONP AND QPN ARE SUPPLEMENTARY. THEREFORE. BY THE TRANSITIVE PROPERTY OF PARALLEL LINES,.

6 11) GIVEN: n m, 1 2 PROVE: p r 1) n m 2) 1 3 3) 1 2 4) 2 3 5) p r 1. GIVEN 2. ALTERNATE INTERIOR ANGLES THEOREM 3. GIVEN 4. TRANSITIVE PROPERTY 5. CONVERSE AIA THEOREM 12) Given: 1 2 Prove: 3 4 1) 1 2 1) Given 2) m 1 + m 3 + m 5 = 180 2) DEFINITION OF STRAIGHT ANGLE 3) m 1 + m = 180 3) SUBSTITUTION PROPERTY 4) m 1 + m 3 = 90 4) SUBTRACTION PROPERTY 5) m 4 + m 2 = m 5 5) VERTICAL ANGLES THOREM 6) m 4 + m 2 = 90 6) SUBSTITUTION PROPERTY 7) m 4 + m 1 = 90 7) SUBSTITUTION PROPERTY (SINCE 1 2) 8) m 1 + m 3 = m 4 + m 1 8) TRANSITIVE PROPERTY 9) m 4 = m 3 9) SUBTRACTION PROPERTY 10) ) DEFINITION OF CONGRUENT

7 13) Write a paragraph proof. Given: a b, a, b m Prove: m PROOF: a ll b and a l means that l b since a line perpendicular to parallel lines is perpendicular to both lines (thm 3-9). Since l b and we are given b m, then l ll m since two lines perpendicular to the same line must be parallel to each other (thm 3-8) 14) Complete the two-column proof. GIVEN: q r PROVE: q r 1.GIVEN VERTICAL ANGLES THEOREM CORRESPONDING ANGLES THEOREM TRANSITIVE PROPERTY

8 15) GIVEN: g h, m 1 =122, m 4 = PROVE: p r 1. g h 2. m 1 =122, m 4 = m 1 = m p r 1) GIVEN 2) GIVEN 3) TRANSITIVE PROPERTY 4) DEFINITION OF CONGRUENT 5) GIVEN 6) TRANSITIVE PROPERTY 7) CONVERSE ALTERNATE INTERIOR ANGLES THM 16) GIVEN: q r, p t PROVE: p t 1) GIVEN 2. l 2 2) ALERNATE EXTERIOR ANGLES THEOREM 3. q r 3) GIVEN ) CORRESPONDING ANGLES THEOREM ) TRANSITIVE PROPERTY

9 17) Write a flow proof Given: 2 and 3 are supplementary. Prove: c ll d 2 & 3 ARE SUPPLEMENTARY GIVEN 1 & 2 ARE SUPPLEMENTARY (LINEAR PAIR) 1 3 ( SUPPLEMENTS THM) c ll d (CONVERSE AEA THM) 18) VERTICAL ANGLES THEOREM GIVEN SAME SIDE INTERIOR ANGLES THEOREM GIVEN ALTERNATE INTERIOR ANGLES THEOREM SUBSTITUTION PROPERTY

10 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. 20) GIVEN: 1 3, 1 and 2 are supplementary PROVE: p r 1. g h 2. 1 and 2 are supplementary and 2 are supplementary 5. p r 1. GIVEN 2. GIVEN 3. GIVEN 4. SUBSTITUTION 5. CONVERSE SSIA THM

11 21) 2 3 (GIVEN) a ll b (CONVERSE AEA THM) 22) Complete the paragraph proof of Theorem 3-8 Given: d ll e, e ll f Prove: d ll f Proof: Because it is given that d ll e, then 1 is supplementary to 2 by the SAME SIDE INTERIOR ANGLES THEOREM. Because it is given that e ll f, then 2 3 by the CORRESPONDING ANGLES THEOEM. Thus, by substitution _ 1 is supplementary to 3 _. And by CONVERSE CORRESPONDING ANGLES THEOREM d ll f.

12 23) VERTICAL ANGLES THEOREM GIVEN CORRESPONDING ANGLES THEOREM SAME SIDE INTERIOR ANGLES THEOREM SUBSTITUTION PROPERTY 24) GIVEN: 1 2, 3 4 PROVE: n p STATEMENTS REASONS ) GIVEN 2. l m 2) CONVERSE CORRESPONDING ANGLES THEOREM ) AIA THEOREM ) GIVEN ) TRANSITIVE PROPERTY 4. n p 4) CONVERSE CORRESPONDING ANGLES THEOREM

13 25) Write a flow proof l ll n (GIVEN) j ll k 12 8 (GIVEN) (CORRESPONDING ANGLES THEOREM) (TRANSITIVE PROPERTY (CONVERSE CAT) 26) PROOF: SINCE WE ARE GIVEN THAT a ll c and b ll c, then a ll b by the TRANSITIVE PROPERTY OF PARALLEL LINES. THUS BY THE ALTERNATE INTERIOR ANGLES THEOREM 1 2. SINCE WE ARE GIVEN m 2 = 65, then m 1 = 65 BY THE DEFINITION OF CONGRUENT.

14 27) Given l 2 Prove QPS and l are right angles 1. l 2 1. GIVEN 2. PS PQ 2. IF SUPPLEMENTARY ANGLES ARE CONGRUENT, THEN THE LINES ARE PERPENDICULAR 3. QPS and 1 are right angles. 3. DEFINITION OF RIGHT ANGLES. 28) GIVEN: j k, 1 2 PROVE: r s 1. j k r s 1.GIVEN 2.CAT (if lines are parallel, then Corrsp are congru) 3.GIVEN 3.TRANSITIVE PROPERTY 5.CONVERSE AIA THM (if alt interior angles are congru, then lines are parall)

15 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, x z. 30) GIVEN: CA ED, m FED = m GCA = 45 PROVE: EF CG 1. CA ED 2. CBE FED 3. m FED = m GCA = FED GCA 4. CBE GCA 5. EF CG 1.GIVEN 2.AIA THM (if lines are parallel, then AIA are congru) 3.GIVEN 3.DEFINITION OF CONGRUENT 3.TRANSITIVE PROPERTY 5.CONVERSE AIA THM (if alt interior angles are congru, then lines are parall)

16 31) Given l 2 Prove 3 and 4 are complementary. PROOF: WE ARE GIVEN THAT l 2. SINCE 1 AND 2 FORM A STRAIGHT ANGLE, m 1 = m 2 = 90. WE ALSO KNOW BY THE VERTICAL ANGLE THEOREM THAT l IS CONGRUENT TO 3 AND 4 COMBINED. THUS m l = m 3 + m 4. USING SUBSTITUTION WE HAVE 90 = m THUS 3 AND 4 ARE COMPLEMENTARY BY THE DEFINITION OF COMPLEMENTARY. 32) Given: m, a b, a Prove: b m 1. m, a b, a 2. 3 IS A RIGHT ANGLE 3. 3 AND 4 ARE SUPPLEMENTARY 4. 4 IS A RIGHT ANGLE 5. b m 1.GIVEN 2. CORRESPONDING ANGLES THEOEM 3.SSIA THM (if lines are parallel, then SSIA are SUPP) 4.DEFINITION OF SUPPLEMENTARY 5.DEFINITION OF PERPENDICULAR

17 33) PROOF: WE ARE GIVEN THAT THUS a ll c BY THE TRANSITIVE PROPERTY OF PARALLEL LINES. WE ARE ALSO GIVEN THAT. IF A LINE IS PERPENDICULAR TO ONE OF TWO PARALLEL LINES, THEN THE LINE IS PERPENDICULAR TO BOTH LINES. THUS. 34) 1. r ll s GIVEN 2. CORRESPONDING ANGLES THEOEM 2. VERTICAL ANGLES THEOREM 2. TRANSITIVE PROPERTY

18 35) Write a flow proof j ll k (GIVEN) 9 3 (AIA THEOREM) m 8 + m 9 = 180 (GIVEN) m 8 + m 3 = 180 (SUBSTITUTION PROPERTY) l ll n (CONVERSE SSIA THM) 36) Complete the paragraph proof of Theorem 3-8 for 3 coplanar lines Proof: 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. Thus by the CONVERSE CORRESPONDING ANGLES THEOREM, l ll m.

19 37) PROOF: WE ARE GIVEN. IF TWO LINES ARE PERPENDICULAR TO THE SAME LINE THEN THE LINES ARE PARALLEL, therefore a ll c. WE ARE ALSO GIVEN THAT c ll d, THUS BY THE TRANSITIVE PROPERTY OF PARALLEL LINES, a ll d. 38) Write a 2-column proof: Given: a b, x y Prove: 4 is supplementary to a b x y is supplementary to is supplementary to 5 1.GIVEN 2. CORRESPONDING ANGLES THEOEM 3. GIVEN 4. CORRESPONDING ANGLES THEOEM 4. TRANSITIVE PROPERTY 5. LINEAR PAIR 5. SUBSTITUTION PROPERTY

20 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 s slope 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 the Converse 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.