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If three or more parallels intercept equal parts on transversal, they intercept equal parts on every transversal.
If three or more parallels intercept equal parts on transversal, they intercept equal parts on every transversal.
A transverse is a line that spans all of the parallel lines.
Draw parallel lines AH, BK, CD, DP through transversal HP such that HK = KM = MP
Draw transversal AD through these lines so that it intercepts lines at A, B, C and D as shown.\
Prove AB = BC = CD
Prove AB = BC = CD
Draw AE, BF and CG parallel to HP such that E, F and G lie on BK, CM and GP respectively
AHKE is a parallelogram ... (Opp. sides of quad. parallel)
∴ AE = HK ... (Opp. side parallelogram are equal)
Similarly BF = KM and CG = MP
But HK = KM = MP ... (Construct)
∴ AE = BF = CG
∠ABE = ∠BCF = ∠CDG ... (Corresponding angles of parallels)
∠HKE = ∠AEB ... (Corresponding angles of parallels)
Similarly ∠KMF = ∠BFC and ∠MPG = ∠CGD
But ∠HKE = ∠KMF = ∠MPG ... (Corresponding angles of parallels)
∴ ∠AEB = ∠BFC = ∠CGD
Consider ∇s AEB, BFC
∠AEB = ∠BFC and ∠ABE = ∠BCF ...(proven above)
∴ ∠BAE = ∠CBF ... (triangle angles equal 180°)
∴ ∇s AEB, BFC and CGD are congruent ... (Two angles and inc. side equal)
∴ AB = BC = CD
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