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CONFIDENTIAL 1
GeometryGeometry
Proving Lines Proving Lines ParallelParallel
CONFIDENTIAL 2
Warm UpWarm Up
Identify each of the following:
1) One pair of parallel segments 2) One pair of skew segments
3) One pair of perpendicular segmentsA
B C
D
E
CONFIDENTIAL 3
Proving lines ParallelProving lines Parallel
The converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it
must be stated as postulate or proved as a separate theorem.
CONFIDENTIAL 4
POSTULATE: If two coplanar lines are cut by a transversal so that a pair of corresponding angles
are congruent, then the two lines are parallel.
Converse of Corresponding angle postulateConverse of Corresponding angle postulate
1 2
1
2
m
n
HYPOTHESIS:
CONCLUSION: m || n
CONFIDENTIAL 5
Using the Converse of Corresponding angle postulateUsing the Converse of Corresponding angle postulate
Use the converse of Corresponding Angles postulate and the given information to show that l || m.
1 2 l
m
3 4
5 6
7 8
/1 = /5 /1 = /5 are Corresponding Angles.
l || m Converse of Corr. /s Angles postulate.
A) /1 = /5
CONFIDENTIAL 6
B) m/4 = (2x + 10)0, m/8 = (3x - 55)0, x = 65
1 2 l
m
3 4
5 6
7 8
m/4 = 2(65) + 10 = 140 Substitute 65 for x.
m/8 = 3(65) – 55 = 140 Substitute 65 for x.
m/4 = m/8 Trans. prop. of equality
/4 = /8 Def. of cong. angles.
l || m Converse of Corr. /s Angles postulate.
CONFIDENTIAL 7
Use the converse of Corresponding Angles postulate and the given information to show that p || q.
Now you try!
1 2 3 45 6 7 8
p q
t
1a) m/1 = m/3
1b) m/7 = (4x + 25)0, m/5 = (5x + 12)0, x = 13
CONFIDENTIAL 8
Through a point P not on line l, there is exactly one line parallel to l.
Parallel postulateParallel postulate
The converse of Corresponding Angles postulate is used to construct parallel lines. The parallel postulate
guarantees that for any line l, you can always construct through a point that is not on l.
CONFIDENTIAL 9
Construction of Parallel linesConstruction of Parallel lines
Draw a line l and a point P not on l.
Draw a line m through P that intersects l. Label the angle 1.
l
P
l
P
l
m1
Construct an angle congruent to /1 at P. By the converse of corresponding angle postulate, l || m.
STEP1:STEP1:
STEP2:
STEP3: l
P
l
m1 2
n
CONFIDENTIAL 10
THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles
are congruent, then the two lines are parallel.
Converse of Alternate interior angles theoremConverse of Alternate interior angles theorem
1
2
m
n
HYPOTHESIS:
CONCLUSION: m || n
Proving lines parallel
CONFIDENTIAL 11
THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles
are congruent, then the two lines are parallel.
Converse of Alternate exterior angles theoremConverse of Alternate exterior angles theorem
3 4
3
4
m
n
HYPOTHESIS:
CONCLUSION: m || n
CONFIDENTIAL 12
THEOREM: If two coplanar lines are cut by a transversal so that a pair of Same side interior angles
are supplementary, then the two lines are parallel.
Converse of Same side interior angles theoremConverse of Same side interior angles theorem
5
6
m
n
HYPOTHESIS:
CONCLUSION: m || n
m/5 = m/6 = 1800
CONFIDENTIAL 13
Converse of Alternate exterior angles theoremConverse of Alternate exterior angles theorem
Proof
1
2
m
n
3
Given: /1 = /2
Prove: l || m
Proof: It is given that /1 = /2. Vertical angles are congruent, so /1 = /3. By the Transitive property of Congruence, /2 = /3. So, l || m by the Converse of Corresponding Angle Postulate.
CONFIDENTIAL 14
1 2 3 48 7 6 5
r s
t
Determining whether lines are parallelDetermining whether lines are parallel
Use the given information and the theorems you have learnt to show that r || s.
/2 = /6 /2 = /6 are Alternate interior Angles.
r || s Converse of Alt. int. Angles theorem.
A) /2 = /6
CONFIDENTIAL 15
B) m/6 = (6x + 18)0, m/7 = (9x + 12)0, x = 10
m/6 = 6(10) + 18 = 78 Substitute 10 for x.
m/7 = 9(65) + 12 = 102 Substitute 10 for x.
m/6 + m/7 = 78 + 12 = 180 /6 and /7 are same-side interior angles.
l || m Converse of same-side interior angles theorem.
1 2 3 48 7 6 5
r s
t
CONFIDENTIAL 16
Proving lines parallelProving lines parallel
Use the given information and the theorems you have learnt to show that r || s.
r || sGiven: l || m, /1 = /3
Prove: r || p
Proof:
l
mm
p
r
1
2
3
Statements Reasons
1. l || m 1. Given
2. /1 = /2 2. corr. Angles Post.
3. /1 = /3 3. Given
4. /2 = /3 4. trans. prop. Of congruency
5. r || p 5. conv. Of Alt ext angles thm.
CONFIDENTIAL 17
Given: /1 = /4, /3 and /4 are supplementary.
Prove: l || m
Now you try!
ll m n
1 23
4
CONFIDENTIAL 18
During a race, all members of a rowing team should keep the oars parallel on each side. If m/1 = (3x + 13)0, m/2 = (5x - 5)0, x
= 9, show that the oars are parallel.
Sports ApplicationSports Application
1
2
A line through the center of the boat forms a transversal to the two oars on each side of the boat.
/1 and /2 are corresponding angles./1 and /2 are corresponding angles.
If /1 = /2, then the oars are parallel.
m/6 = 3(9) + 13 = 40
m/7 = 5(9) - 5 = 40
Substitute 10 for x in each expression.
m/1 = m/2, /1 = /2.
The corresponding angles are congruent, so the oars are parallel by the converse of corresponding angles postulates.
CONFIDENTIAL 19
Now you try!
1
2
4) Suppose the corresponding angles on the opposite side of the boat measure (4y - 2)0 and (3y + 6)0, where y = 8.
Show that the oars are parallel.
CONFIDENTIAL 20
Assessment
Use the converse of Corresponding Angles postulate and the given information to show that p || q.
1 2 8 74 3 5 6
p q
t
1) m/4 = m/5
2) m/1 = (4x + 16)0, m/8 = (5x - 12)0, x = 28
3) m/4 = (6x - 19)0, m/5 = (3x + 14)0, x = 11
CONFIDENTIAL 21
Use the given information and the theorems you have learnt to show that r || s.
12 3
87 6
54
rs
4) m/1 = m/5
5) m/3 + m/4 = 1800
6) m/3 = m/7
7) m/4 = (13x - 4)0, m/8 = (9x + 16)0, x = 5
8) m/8 = (17x + 37)0, m/7 = (9x - 13)0, x = 6
9) m/2 = (25x + 7)0, m/6 = (24x + 12)0, x = 5
CONFIDENTIAL 22
10) Complete the following 2 column proof:
XX
Y V
W
2
1 3Given: /1 = /2, /3 = /1
Prove: XY || VW
Proof:
Statements Reasons
1. /1 = /2, /3 = /1 1. Given
2. /2 = /3 2. a._______
3.b. ______ 3. c._______
CONFIDENTIAL 23
POSTULATE: If two coplanar lines are cut by a transversal so that a pair of corresponding angles
are congruent, then the two lines are parallel.
Converse of Corresponding angle postulateConverse of Corresponding angle postulate
1 2
1
2
m
n
HYPOTHESIS:
CONCLUSION: m || n
Let’s review
CONFIDENTIAL 24
Using the Converse of Corresponding angle postulateUsing the Converse of Corresponding angle postulate
Use the converse of Corresponding Angles postulate and the given information to show that l || m.
1 2 l
m
3 4
5 6
7 8
/1 = /5 /1 = /5 are Corresponding Angles.
l || m Converse of Corr. /s Angles postulate.
A) /1 = /5
CONFIDENTIAL 25
B) m/4 = (2x + 10)0, m/8 = (3x - 55)0, x = 65
1 2 l
m
3 4
5 6
7 8
m/4 = 2(65) + 10 = 140 Substitute 65 for x.
m/8 = 3(65) – 55 = 140 Substitute 65 for x.
m/4 = m/8 Trans. prop. of equality
/4 = /8 Def. of cong. angles.
l || m Converse of Corr. /s Angles postulate.
CONFIDENTIAL 26
Through a point P not on line l, there is exactly one line parallel to l.
Parallel postulateParallel postulate
The converse of Corresponding Angles postulate is used to construct parallel lines. The parallel postulate
guarantees that for any line l, you can always construct through a point that is not on l.
CONFIDENTIAL 27
Construction of Parallel linesConstruction of Parallel lines
Draw a line l and a point P not on l.
Draw a line m through P that intersects l. Label the angle 1.
l
P
l
P
l
m1
Construct an angle congruent to /1 at P. By the converse of corresponding angle postulate, l || m.
STEP1:STEP1:
STEP2:
STEP3: l
P
l
m1 2
n
CONFIDENTIAL 28
THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles
are congruent, then the two lines are parallel.
Converse of Alternate interior angles theoremConverse of Alternate interior angles theorem
1
2
m
n
HYPOTHESIS:
CONCLUSION: m || n
Proving lines parallel
CONFIDENTIAL 29
THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles
are congruent, then the two lines are parallel.
Converse of Alternate exterior angles theoremConverse of Alternate exterior angles theorem
3 4
3
4
m
n
HYPOTHESIS:
CONCLUSION: m || n
CONFIDENTIAL 30
THEOREM: If two coplanar lines are cut by a transversal so that a pair of Same side interior angles
are supplementary, then the two lines are parallel.
Converse of Same side interior angles theoremConverse of Same side interior angles theorem
5
6
m
n
HYPOTHESIS:
CONCLUSION: m || n
m/5 = m/6 = 1800
CONFIDENTIAL 31
Converse of Alternate exterior angles theoremConverse of Alternate exterior angles theorem
Proof
1
2
m
n
3
Given: /1 = /2
Prove: l || m
Proof: It is given that /1 = /2. Vertical angles are congruent, so /1 = /3. By the Transitive property of Congruence, /2 = /3. So, l || m by the Converse of Corresponding Angle Postulate.
CONFIDENTIAL 32
Proving lines parallelProving lines parallel
Use the given information and the theorems you have learnt to show that r || s.
r || sGiven: l || m, /1 = /3
Prove: r || p
Proof:
l
mm
p
r
1
2
3
Statements Reasons
1. l || m 1. Given
2. /1 = /2 2. corr. Angles Post.
3. /1 = /3 3. Given
4. /2 = /3 4. trans. prop. Of congruency
5. r || p 5. conv. Of Alt ext angles thm.
CONFIDENTIAL 33
You did a great job You did a great job today!today!
