1) Given: 1 and 4 are supplementary.
Prove: a b
1 and 4 are
supplementary
2 and 3 are supplementary
GIVEN
Substitution Property
1  2 and
3 4
a ll b
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_____.
ANGLES__THM____.
This means that 1   2 because the ___CORRESPONDING
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__________.
3) GIVEN: g || h, 1  2
PROVE: p || r
4)
Statements
1) g || h
Reasons
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
Given:
m, a b
Prove: 1  5
Statements
Reasons
1. Given
1.
m, a b
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
7.
4  5
7. VERTICAL ANGLES THEOREM (VAT)
8.
1  5
8. TRANSITIVE PROPERTY
6. TRANSITIVE PROPERTY
5) Given: 1 and 2 are supplementary; x ║ y
Prove: q ║ r
x ll y
GIVEN
2 and 3 are
supplementary
SSIA THEOREM
1  3
 SUPPLEMENTS THM
1 and 2 are
supplementary
q ll r
.
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__________.
7) GIVEN: p  q, q || r
PROVE: p  r
Statements
1. p  q
Reasons
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
1. g || h
Reasons
1. GIVEN
2.  1   3
2. CORRESPONDING ANGLES THOREM (CAT)
3.  1   2
3. GIVEN
4.  2   3
4. TRANSITIVE PROPERTY
5. p || r
5. CONVERSE CORRESPONDING ANGLES THEOREM
9) Given: 1 is supplementary to 2
Prove:
1
l
m
m
2
3
1 and 2 are
supplementary
GIVEN
1  3
l ll m
Congruent Supplements
Theorem
2 and 3 are
supplementary
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 ⃡
⃡
ONP AND QPN ARE SUPPLEMENTARY. THEREFORE ⃡
TRANSITIVE PROPERTY OF PARALLEL LINES, ⃡
⃡
AND ⃡
⃡
.
⃡
. BY THE
THUS
11) GIVEN: n || m, 1  2
PROVE: p || r
Statements
1) n || m
Reasons
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
1) 1  2
Reasons
1) Given
2) m1 + m3 + m5 = 180
2) DEFINITION OF STRAIGHT ANGLE
3) m1 + m3 + 90 = 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) 3  4
10) DEFINITION OF CONGRUENT
13) Write a paragraph proof.
Given: a b , a  , b  m
Prove:
PROOF:
m
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
1. q || r
Reasons
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, m1 =122, m4 = 122 1  3
PROVE: p || r
Statements
1. g || h
2. m1 =122, m4 = 122
3. m1 = m4
Reasons
1) GIVEN
2) GIVEN
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
5.  1   3
5) TRANSITIVE PROPERTY
17) Write a flow proof
Given: 2 and 3 are supplementary.
Prove: c ll d
2 & 3 ARE SUPPLEMENTARY
GIVEN
1 & 2 ARE SUPPLEMENTARY
c ll d
1  3
( SUPPLEMENTS
THM)
(CONVERSE AEA
THM)
(LINEAR PAIR)
18)
VERTICAL ANGLES THEOREM
GIVEN
SAME SIDE INTERIOR ANGLES THEOREM
GIVEN
ALTERNATE INTERIOR ANGLES THEOREM
SUBSTITUTION PROPERTY
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
1. g || h
Reasons
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)
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.
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
1.  1   2
2. l ║ m
3. 4   5
3. 3   4
4. 3   5
4. n║ p
1) GIVEN
2) CONVERSE CORRESPONDING ANGLES THEOREM
3) AIA THEOREM
3) GIVEN
4) TRANSITIVE PROPERTY
4) CONVERSE CORRESPONDING ANGLES THEOREM
25) Write a flow proof
l ll n
(GIVEN)
8  4
12  4
12  8
(CORRESPONDING
ANGLES
THEOREM)
(CORRESPONDING
ANGLES
THEOREM)
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 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.
27) Given l  2
Prove QPS and l are right angles
28)
Statements
1. l  2
Reasons
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.
GIVEN: j║ k,  1   2
PROVE: r║ s
Statements
1. j || k
Reasons
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.CONVERSE AIA THM (if alt interior angles are
5. r || s
congru, 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, ___x
z______.
30) GIVEN: CA ║ ED ,
m FED = m GCA = 45°
PROVE: EF ║ CG
Statements
1. CA ║ ED
2. CBE  FED
3. m FED = m GCA = 45°
Reasons
1.GIVEN
2.AIA THM (if lines are parallel, then AIA are congru)
3.GIVEN
3.  FED   GCA
3.DEFINITION OF CONGRUENT
4.  CBE   GCA
3.TRANSITIVE PROPERTY
5.CONVERSE AIA THM (if alt interior angles are
5. EF ║ CG
congru, then lines are parall)
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3 + 4. THUS 3 AND 4 ARE COMPLEMENTARY
BY THE DEFINITION OF COMPLEMENTARY.
32) Given:
m, a b, a 
Prove: b  m
Statements
1.
m, a b, a 
Reasons
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
1. r ll s
Reasons
1.GIVEN
2.  1  6
2. CORRESPONDING ANGLES THEOEM
2.  8  6
2. VERTICAL ANGLES THEOREM
2.  1  8
2. TRANSITIVE PROPERTY
35) Write a flow proof
j ll k
(GIVEN)
m8 + m9 = 180
(GIVEN)
9  3
(AIA THEOREM)
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.
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
1. a ║ b
Reasons
1.GIVEN
2.  4  12
2. CORRESPONDING ANGLES THEOEM
3. x ║ y
3. GIVEN
4.  12  16
4. CORRESPONDING ANGLES THEOEM
5.  4  16
4. TRANSITIVE PROPERTY
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.