Intercept
In the figure given above, XY is a transversal cutting the line L_{1} and L_{2} at P and Q respectively. The line segment PQ is called the intercept made on the transversal XY by the lines L_{1} and L_{2}.
If a transversal makes equal intercepts on three or more parallel lines then any other transversal cutting them will also make equal intercepts.
Given: Let there be three straight lines L_{1}, L_{2}, and L_{3} such that L_{1} ∥ L_{2} ∥ L_{3}.
Transversal AB makes equal intercepts on L1, L2 and L3, I.e., PQ = QR. Another transversal CD makes intercepts KM and MN.
To Prove: KM = MN.
Construction: Join PN which cuts the L_{2} at O.
Proof:
Statement |
Reason |
1. PQ = QR and QO ∥ line L3. |
1. Given. |
2. O is the midpoint of PN, i.e., PO = ON. |
2. By converse of Midpoint Theorem. |
3. PO = ON and OM ∥ L1. |
3. By statement 2 and given. |
4. M is the midpoint of NK, i.e., KM = MN (Proved) |
4. By converse of Midpoint Theorem. |
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