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 = 1800, BGH+DHG =1800

 

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= 1800

Enunciation 1 : The ray FE is standing  on the straight line AB

8

BGH = CHG

From Step 6

9

AGH+CHG= 1800

Substitute CHG for HGB in step 7

10

CHG +GHD = 1800

Enunciation1 : The ray FE is standing  on the straight line CD

11

BGH = CHG

From step 6

12

GHD+BGH= 1800

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, AB||CD. 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 = 2FEG

Given that EH bisects FEB

3

CFE = 2FEG

Equality of Step 1 and 2

4

EFD = AEF

Alternate  angles: AB ||CD

5

EFD = 2EFG

Given that GF bisects EFD

6

CFE +EFD = 1800

Angles on a straight line CD

7

2FEG +2EFG = 1800

Substitute 3 and 5 in 6

8

FEG +EFG = 900

Simplification of 7

9

FEG = GEB

EG bisects BEI

10

GEB = EGI

Alternate angles AB||IG

11

FEG = EGI

Equate 9 and 10

12

EFG=GFD

FG bisects IFD

13

GFD =IGF

Alternate angles CD||IG

14

EFG =IGF

Equate 12 and 13

15

EGI +IGF(=EGF) = 900

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 = 1800 (BGH +DHG = 1800)

 

TO prove: AB||CD.

 

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.