6.3 Theorem
on Parallel lines:
‘Theorem’ is a proposition in
which some statements are to be proved logically.
Theorem has following parts;
1. Data(Hypothesis)  Lists the facts given in the theorem.
2. Figure relevant for the theorem
3. To prove The statement or proposition which is
to be proved
4. Construction if any(More
details added to the figure drawn in
step 2)
5. Proof (Series of several steps)
An example of a theorem which will be
proved later is:
Pythagoras’s theorem: Square of hypotenuse in a right angled triangle is equal to sum
of squares of other two sides (Hypotenuse)^{2} = (Side)^{2 } + (Side)^{2}^{} 

Theorem can not be proved by just by giving several
examples. It needs to be proved by one of the following two methods:
1. Based on axioms or theorems already proved.
2. Some theorems are proved by negation (we start
with the assumption that theorem is wrong and then arrive at a contradiction or
mathematical absurdity. This forces
us to arrive at a conclusion that our assumption was wrong and hence theorem
must be true).
However, to disprove a statement, an
example which does not satisfy the given statement will be enough.
6.3 Theorem 1: If a transversal line cuts two parallel lines then
1) Each pair of alternate angles is equal
2) The interior angles on the same side of the
transversal are supplementary
Data: AB  CD, Transversal EF cuts AB at G and CD
at H
To prove:
1) _{}AGH = _{}GHD, _{}BGH=_{}CHG
2) _{}AGH+_{}CHG = 180^{0}, _{}BGH+_{}DHG =180^{0}
No 
Statement 
Reason 

1 
_{}EGB = _{}GHD 
Enunciation
3 : Corresponding angles are equal when a transversal cuts parallel lines 

2 
_{}EGB = _{}AGH 
Enunciation
2 : vertically opposite angles are equal 

3 
_{}AGH = _{}GHD 
Axiom
1 for angles in steps 1 and 2 Things
which are equal to the same thing are equal to each other 

4 
_{}AGE =_{}CHG 
Enunciation
3: Corresponding angles 

5 
_{}AGE = _{}BGH 
Enunciation
2: Vertically opposite angle 

6 
_{}CHG=_{}BGH 
Axiom
1 for angles in step 4 and 5 

7 
_{}AGH+_{}HGB= 180^{0} 
Enunciation
1 : The ray FE is standing on the
straight line AB 

8 
_{}BGH = _{}CHG 
From
Step 6 

9 
_{}AGH+_{}CHG= 180^{0} 
Substitute
_{}CHG for _{}HGB in step 7 

10 
_{}CHG +_{}GHD = 180^{0} 
Enunciation1
: The ray FE is standing on the
straight line CD 

11 
_{}BGH = _{}CHG 
From
step 6 

12 
_{}GHD+_{}BGH= 180^{0} 
Substitute
_{}BGH for _{}CHG in step 10 
6.3 Problem 1: In the figure AB  PQ and BC  QR. Prove that _{}PQR =_{}ABC
Data: AB 
PQ and BC  QR
To Prove: _{}PQR =_{}ABC
Construction:
Extend PQ to cut BC at T, Extend QR to cut AB at S
Proof:
_{}PQR = _{}ASR (corresponding angles) _{}ASR = _{}ABC (corresponding angles) _{}_{}PQR =_{}ABC 

6.3 Problem 2: In the adjacent
figure, ABCD. EH and FG are the angular bisectors of _{}FEB and _{}EFD respectively.
Prove that EH and FG are
perpendicular to each other.
Construction: Draw GI
parallel to CD passing through G
Solution:
No 
Statement 
Reason 

1 
_{}CFE = _{}BEF 
Alternate angles: AB CD 

2 
_{}BEF = 2_{}FEG 
Given that EH bisects _{}FEB 

3 
_{}CFE = 2_{}FEG 
Equality of Step 1 and 2 

4 
_{}EFD = _{}AEF 
Alternate angles: AB CD 

5 
_{}EFD = 2_{}EFG 
Given that GF bisects _{}EFD 

6 
_{}CFE +_{}EFD = 180^{0} 
Angles on a straight line CD 

7 
2_{}FEG +2_{}EFG = 180^{0} 
Substitute 3 and 5 in 6 

8 
_{}FEG +_{}EFG = 90^{0} 
Simplification of 7 

9 
_{}FEG = _{}GEB 
EG bisects _{}BEI 

10 
_{}GEB = _{}EGI 
Alternate angles ABIG 

11 
_{}FEG = _{}EGI 
Equate 9 and 10 

12 
_{}EFG=_{}GFD 
FG bisects _{}IFD 

13 
_{}GFD =_{}IGF 
Alternate angles CDIG 

14 
_{}EFG =_{}IGF 
Equate 12 and 13 

15 
_{}EGI +_{}IGF(=_{}EGF) = 90^{0} 
Substituting 11 and 14 in 8 

16 
Thus EH and FG are perpendicular
to each other 
6.3 Theorem 2(Converse of the theorem 6.3.1): If a transversal line
cuts two straight lines such that
Case1): Each pair of alternate angles is equal
OR
Case 2): The interior angles on the same side of
the transversal are supplementary
Then the straight lines are parallel.
Given:
1) Transversal EF cuts two straight lines AB and CD
at G and H respectively. And
2) _{}AGH = _{}GHD (_{}BGH=_{}CHG)
OR
3) _{}AGH +_{}CHG = 180^{0 }(_{}BGH +_{}DHG = 180^{0})
TO prove: ABCD.
Hint: In
both the cases show that the corresponding angles are equal and then use the
enunciation ‘6.1.3 Enunciation 4’ to show that these lines are parallel 

6.3 Summary of learning
No 
Points to remember 
1 
If
a transversal line cuts two parallel lines then 1)
Each pair of alternate angles are equal 2)
The interior angles on the same side of the transversal are supplementary 
2 
If
a transversal line cuts two straight lines such that Case
1): Each pair of alternate angles is equal OR Case
2): The interior angles on the same side of the transversal are supplementary
Then
the straight lines are parallel. 