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Equal Intercepts Theorem

Intercept

Intercept

In the figure given above, XY is a transversal cutting the line L1 and L2 at P and Q respectively. The line segment PQ is called the intercept made on the transversal XY by the lines L1 and L2.

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 L1, L2, and L3 such that L1 ∥ L2 ∥ L3.

Transversal makes Equal Intercepts

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 L2 at O.

Equal Intercepts Theorem

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.





9th Grade Math

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