1) Given: 1 and 4 are supplementary. Prove · 13) Write a paragraph proof. Given: a b a b m,,...
Transcript of 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)
8. 1 5 8. TRANSITIVE PROPERTY
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
4. 2 3 4. TRANSITIVE PROPERTY
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
5. 1 3 5) TRANSITIVE PROPERTY
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. 3 5 4) TRANSITIVE PROPERTY
4. n║ p 4) CONVERSE CORRESPONDING ANGLES THEOREM
VERTICAL ANGLES THEOREM
GIVEN
CORRESPONDING ANGLES THEOREM
SAME SIDE INTERIOR ANGLES THEOREM
SUBSTITUTION PROPERTY
25) Write a flow proof
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
2. 1 8 2. TRANSITIVE PROPERTY
35) Write a flow proof
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
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.