Q61 of 93 Page 6

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.
                  


Given: AB and CD are two lines cut by transversal EF.
    QP is the bisector of the angle ∠EQB. SR is the bisector of the corresponding ∠QSD.
    PQ || SR.
To prove: AB || CD
Proof: QP || SR and are cut by the transversal EF.
     എQP = ∠QSR        (Corresponding angles are equal.)------ (1)
PQ is the bisector of ∠EQB
... ∠EQP = ∠PQB = (1/2) ∠EQB     ------------------(2)
Similarly, SR is the bisector of ∠QSD
   ∠QSR = ∠RSD = (1/2) ∠QSD  ------------------(3)
But, from (1)
   ∠EQP = ∠QSR
⇒  (1/2) ∠EQB = (1/2) ∠QSD (From 2 and 3)
... ∠EQB = ∠QSD.
AB and CD are two lines cut by a transversal EF such that corresponding angles.
   ∠EQB = ∠QSD
... AB || CD (If a transversal intersects two lines such that a pair of corresponding angles are equal then the lines are parallel.)

More from this chapter

All 93 →
59 If two parallel lines are intersected by a transversal, then prove that the bisectors of any two alternate angles are parallel.
                           60 In figure, bisectors GM and HL of alternate angles AGH and DHG respectively are parallel to each other. Prove that AB || CD.
                   62 A transversal intersects two given lines in such a way that the interior angles on the same side of  the transversal are equal. Is it always true that the given lines are parallel? If not, state the condition (s) under which the two lines will be parallel. 63 In figure, arms BA and BC of ∠ ABC are respectively parallel to arms ED and EF of ∠ DEF. Prove that ∠ ABC =∠ DEF.