Q59 of 93 Page 6

If two parallel lines are intersected by a transversal, then prove that the bisectors of any two alternate angles are parallel.
                          


Given: AB and CD are 2 parallel intersected by a transversal EF. PO and QR are the bisectors of the alternate angles ∠ APQ and ∠ PQD respectively. To prove: OP || QR. Proof: Since AB || CD they cut by a transversal EF. ⇒ ∠ APQ = ∠ PQD (Alternate angles are equal)----------- (1)OP is the bisector of ∠ APQ  \\ ∠ APO = ∠ OPQ or, ∠ APO = ∠ APQ -------------- (2)Similarly, QR is the bisector of ∠ PQD ∠ PQR = ∠ RQD = ∠ PQD ………….. (3)From (1)   ∠ APQ = ∠ PQD ∠ APQ = ∠ PQD (From 2 and 3)   Ð OPQ = ∠ PQR Since, OP and QR are two lines cut by transversal EF such ∠ OPQ = ∠ PQR. ∴ OP || QR (If a transversal intersects two lines in such a way that a pair of alternate angles are equal then the two lines are parallel.) ∴ The bisectors of any pair of alternate interior angles are parallel.

More from this chapter

All 93 →
57 If p, m, n are three lines such that p || m and n ⊥ p, prove that n ⊥ m. 58 In figure, EF is a transversal to two parallel lines AB and CD. GM ad HL are the bisectors of the corresponding angles EGB and EHD. Prove that GM||HL.
                   60 In figure, bisectors GM and HL of alternate angles AGH and DHG respectively are parallel to each other. Prove that AB || CD.
                   61 If two lines are intersected by a transversal in such a way that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.