Question

If three or more parallel lines are intersected by two transversals , prove that the intercepta made by them on the transversals are proportional.

Answer 1

Let the three parallel straight lines AB, CD, EF make equal intercepts KL and LM from the transversal IJ, that is KL = LM.
The intercepts made by these three parallel lines on the transversal XY are PQ and QR.
Construction: Through Q, a straight line is drawn parallel to IJ to intersect AB and EF at U and V respectively.

Proof:

For quadrilateral KLQU,
KU ∥ LQ       [∵, AB ∥ CD] and KL ∥ UQ       [By construction]
∴ KLQU is a parallelogram.
∴ KL = UQ
Similarly, from quadrilateral LMVQ, we get LM = QV.
But it is given that KL = LM.
∴ UQ = QV.
Now, from △UPQ and △QVR, we get
∠PUQ = alternate ∠QVR       [∵ AB ∥ EF, UV is the transversal]
∠PQU = vertically opposite ∠VQR
UQ = QV       [Proved before]
∴ △UPQ ≅ △VRQ       [By A-A-S condition of congruence]
∴ PQ = QR       [Corresponding sides of two congruent triangles]
Thus the theorem is proved for three parallel straight line.

Given, KL = LM = MN
To prove, PQ = QR = RS.
By drawing a straight line through Q, parallel to IJ, we have proved that PO = QR.
Again, a straight line is drawn through R parallel to IJ to intersect CD and GH at Z and W a respectively.
As before, if can be proved that QR = RS.
∴ PQ = QR = RS
In this way, the theorem can be proved for any number of parallel straight lines greater than 3.
Remark: From figure 2, we get:
KL = LM = MN implies PQ = QR =RS.
∴ L is the mid point of KM.
That is, KM = 2KL.
∴ KMKL=21
or, KMMN=21       [∵ KL = MN],
∴ KM : MN = 2 : 1
Similarly from PQ = QR = RS, we get PR : RS = 2: 1
So, it can be said:

If three parallel straight lines make two intercepts from a transversal in the ratio 2 : 1, then those three parallel straight lines will make two intercepts from transversal in the ratio 2 : 1.
Likewise, in an essentially similar manner, it can be shown that if three or more parallel straight lines make intercepts from a transversal in a certain ratio, then those parallel straight lines will make intercepts from any other transversal in the same ratio.