Question

LE 15 If three or more parallel lines are intersected by two transversa
prove that the intercepts made by them on the transversas
proportional
plz make it fast​

Answer 1

Answer:

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.