1) given: 1 and 4 are supplementary. prove · 13) write a paragraph proof. given: a b a b m,,...

1) Given: 1 and 4 are supplementary.

Prove: a b

GIVEN

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, m1 = 90. So, m2 =_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__________.

1 and 4 are

supplementary

1 2 and

3 4

2 and 3 are supplementary a ll b

Substitution Property

3) GIVEN: g || h, 1 2

PROVE: p || r

Statements Reasons

1) g || h 1. GIVEN

2) 1 3 2. CORRESPONDING ANGLES THEOREM (CAT)

3) 1 2 3. GIVEN

4) 2 3 4. TRANSITIVE PROPERTY

5) p || r 5. CONVERSE AEA THEOREM

4) Given: ,m a b

Prove: 1 5

Statements Reasons

1. ,m a b 1. Given

2. 1 2 2. 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 4 5. CONGRUENT SUPPLEMENTS THEOREM (IF TWO

ANGLES ARE SUPPLEMENTARY TO THE SAME ANGLE THOSE

ANGLES ARE CONGRUENT)

6. 1 4 6. TRANSITIVE PROPERTY

7. 4 5 7. VERTICAL ANGLES THEOREM (VAT)


5) Given: 1 and 2 are supplementary; x ║ y

Prove: q ║ r

GIVEN SSIA THEOREM

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__________.

x ll y

2 and 3 are

supplementary

1 3 q ll r

. 1 and 2 are

supplementary

7) GIVEN: p q, q || r

PROVE: p r

Statements Reasons

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

5. 1 2 5) 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

Statements Reasons

1. g || h 1. GIVEN

2. 1 3 2. CORRESPONDING ANGLES THOREM (CAT)

3. 1 2 3. GIVEN


5. p || r 5. CONVERSE CORRESPONDING ANGLES THEOREM

9) Given: 1 is supplementary to 2

Prove: m

GIVEN

CONVERSE AEA THM

LINEAR PAIR

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, ⃡ ⃡ .

1

3

2

l

m

1 and 2 are

supplementary

1 3 l ll m

2 and 3 are

supplementary

Congruent Supplements

Theorem

11) GIVEN: n || m, 1 2

PROVE: p || r

Statements Reasons

1) n || m 1. GIVEN

2) 1 3 2. ALTERNATE INTERIOR ANGLES THEOREM

3) 1 2 3. GIVEN

4) 2 3 4. TRANSITIVE PROPERTY

5) p || r 5. CONVERSE AIA THEOREM

12) Given: 1 2

Prove: 3 4

Statements Reasons

1) 1 2 1) Given

2) m1 + m3 + m5 = 180 2) DEFINITION OF STRAIGHT ANGLE

3) m1 + m3 + 90 = 180 3) SUBSTITUTION PROPERTY

4) m1 + m3 = 90 4) SUBTRACTION PROPERTY

5) m4 + m2 = m5 5) VERTICAL ANGLES THOREM

6) m4 + m2 = 90 6) SUBSTITUTION PROPERTY

7) m4 + m1 = 90 7) SUBSTITUTION PROPERTY (SINCE 1 2 )

8) m1 + m3 = m4 + m1 8) TRANSITIVE PROPERTY

9) m4 = m3 9) SUBTRACTION PROPERTY

10) 3 4 10) DEFINITION OF CONGRUENT

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: 1 3

Statements Reasons

1. q || r 1.GIVEN

2. 1 2 2.VERTICAL ANGLES THEOREM

3. 2 3 3.CORRESPONDING ANGLES THEOREM

4. 1 3 4.TRANSITIVE PROPERTY

15) GIVEN: g || h, m1 =122, m4 = 122 1 3

PROVE: p || r

Statements Reasons

1. g || h 1) GIVEN

2. m1 =122, m4 = 122 2) GIVEN

3. m1 = m4 3) TRANSITIVE PROPERTY

4. 1 4 4) DEFINITION OF CONGRUENT

5. 1 3 5) GIVEN

6. 3 4 6) TRANSITIVE PROPERTY

7. p || r 7) CONVERSE ALTERNATE INTERIOR ANGLES THM

16) GIVEN: q || r, p || t

PROVE: 1 3

Statements Reasons

1. p || t 1) GIVEN

2. l 2 2) ALERNATE EXTERIOR ANGLES THEOREM

3. q || r 3) GIVEN

4. 2 3 4) CORRESPONDING ANGLES THEOREM


17) Write a flow proof

Given: 2 and 3 are supplementary. Prove: c ll d

18)

VERTICAL ANGLES THEOREM

GIVEN

SAME SIDE INTERIOR ANGLES THEOREM

GIVEN

ALTERNATE INTERIOR ANGLES THEOREM

SUBSTITUTION PROPERTY

GIVEN

(LINEAR PAIR)

( SUPPLEMENTS

THM)

(CONVERSE AEA

THM) 1 & 2 ARE SUPPLEMENTARY

2 & 3 ARE SUPPLEMENTARY

1 3 c ll d

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

Statements Reasons

1. g || h 1. GIVEN

2. 1 and 2 are supplementary 2. GIVEN

3. 1 3 3. GIVEN

4. 3 and 2 are supplementary 4. SUBSTITUTION

5. p || r 5. CONVERSE SSIA THM

21)

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 23 by the

__CORRESPONDING ANGLES THEOEM. Thus, by substitution _1 is supplementary to 3 _.

And by ___CONVERSE CORRESPONDING ANGLES THEOREM d ll f.

(GIVEN) (CONVERSE AEA

THM)

2 3 a ll b

23)

24)

GIVEN: 1 2, 3 4

PROVE: n║ p

STATEMENTS REASONS

1. 1 2 1) GIVEN

2. l ║ m 2) CONVERSE CORRESPONDING ANGLES THEOREM

3. 4 5 3) AIA THEOREM

3. 3 4 3) GIVEN


4. n║ p 4) CONVERSE CORRESPONDING ANGLES THEOREM

VERTICAL ANGLES THEOREM

GIVEN

CORRESPONDING ANGLES THEOREM

SAME SIDE INTERIOR ANGLES THEOREM

SUBSTITUTION PROPERTY


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 m2 = 65, then m1 = 65 BY THE

DEFINITION OF CONGRUENT.

(GIVEN)

(GIVEN)

(CORRESPONDING

ANGLES

THEOREM)

(CONVERSE CAT)

12 8

l ll n

8 4 j ll k

(TRANSITIVE

PROPERTY

12 4

27) Given l 2

Prove QPS and l are right angles

Statements Reasons

1. l 2 1. GIVEN

2. 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

Statements Reasons

1. j || k 1.GIVEN

2. 1 5 2.CAT (if lines are parallel, then Corrsp are congru)

3. 1 2 3.GIVEN

4. 5 2 3.TRANSITIVE PROPERTY

5. r || s

5.CONVERSE AIA THM (if alt interior angles are

congru, then lines are parall)

PS PQ

29) Complete the paragraph proof of the Perpendicular Transversal Theorem (Thm 3-10)

Proof: Since y ll z, m1 = _90__ by the __CORRESPONDING ANGLES THEOREM___.

By definition of _PERPENDICULAR___lines, ___x z______.

30) GIVEN: ║ ,

m FED = m GCA = 45°

PROVE: ║

Statements Reasons

1. ║ 1.GIVEN

2. CBE FED 2.AIA THM (if lines are parallel, then AIA are congru)

3. m FED = m GCA = 45° 3.GIVEN

3. FED GCA 3.DEFINITION OF CONGRUENT

4. CBE GCA 3.TRANSITIVE PROPERTY

5. ║

5.CONVERSE AIA THM (if alt interior angles are

congru, then lines are parall)

CA ED

EF CG

CA ED

EF CG

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, m1 = m2 = 90°. WE ALSO KNOW BY THE VERTICAL ANGLE THEOREM

THAT l IS CONGRUENT TO 3 AND 4 COMBINED. THUS ml = m3 + m4. USING

SUBSTITUTION WE HAVE 90° = m3 + 4. THUS 3 AND 4 ARE COMPLEMENTARY

BY THE DEFINITION OF COMPLEMENTARY.

32) Given: , ,m a b a

Prove: b m

Statements Reasons

1. , ,m a b a 1.GIVEN

2. 3 IS A RIGHT ANGLE 2. CORRESPONDING ANGLES THEOEM

3. 3 AND 4 ARE

SUPPLEMENTARY

3.SSIA THM (if lines are parallel, then SSIA are SUPP)

4. 4 IS A RIGHT ANGLE 4.DEFINITION OF SUPPLEMENTARY

5. b m 5.DEFINITION OF PERPENDICULAR

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)

Statements Reasons

1. r ll s

1.GIVEN

2. 1 6 2. CORRESPONDING ANGLES THEOEM

2. 8 6 2. VERTICAL ANGLES THEOREM



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.

(SUBSTITUTION

PROPERTY)

m8 + m3 = 180

(GIVEN)

(GIVEN)

(AIA THEOREM)

(CONVERSE SSIA

THM)

m8 + m9 = 180

j ll k 9 3

l ll n

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 15

Statements Reasons

1. a ║ b 1.GIVEN


3. x ║ y 3. GIVEN



6. 16 is supplementary to 5 5. LINEAR PAIR

7. 4 is supplementary to 5 5. 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’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.

1) given: 1 and 4 are supplementary. prove · 13) write a paragraph proof. given: a b a b m,,...

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