1. If two lines intersect, prove that the vertically opposite angles are equal.2. Bisectors of interior B and exterior ACD of a Δ ABC intersect at the point T.Prove that BTC =12 BAC.3. A transversal intersects two parallel lines. Prove that the bisectors of any pair ofcorresponding angles so formed are parallel.

1) 

the st. lines AB and CD intersect in O.

therefore, angle AOC+angle AOD = 180 ...(linear pair)

angle AOD+angle BOD = 180.....(linear pair)

from (i) and (ii), we get,

angle AOC+angle AOD= angle AOD +angle BOD

angle AOC = angle BOD 

similarly,

angle BOC = angle AOD

thus, if two straight intersect each other , then the pair of vert. opp. angles are equal.

3)

Given: A transversal EF cuts two parallel lines AB and CD at point G and H resp. 

GL and HM are the bisectors of a pair of a corresponding angles EGB and GHD resp.

To prove: GL is parallel to HM

Proof: since AB is parallel CD and EF is a transversal