Understanding the Intercept Theorem

The Intercept Theorem Watch Video

The theorem states if a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.
Given AP is parallel to BQ is parallel to CR. The intercepts AB, BC and CD are equal. PQ, QR and RS are intercepts on any other line.
To prove that PQ is equal to QR
Construction Through A and B, draw AE and BF parallel to PQR .
Click GO for the proof of the theorem
Let us compare triangle ABE and triangle BCF
Angle ABE is equal to angle BCF since they are the corresponding angles of parallel lines BQ and CR
Angle BAE is equal to angle CBF since they are the corresponding angles of parallel lines AE and BF
AB is equal to BC this data is given to us
Therefore, by AAS congruency triangle ABE is congruent to triangle BCF
Hence AE is equal to BF
AP is parallel to EQ and AE is parallel to PQ, hence APQE is a parallelogram.
Therefore, AE is equal to PQ
Similarly, BQ is parallel to FR and BF is parallel to QR, hence BQRF is a parallelogram
Therefore, BF is equal to QR . BF is equal to QR is equal to AE is equal to PQ
Therefore, PQ is equal to QR.