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.