Q. 104.4( 50 Votes )

In figure 3.13, line DE || line GF ray EG and ray FG are bisectors of DEF and DFM respectively.

Prove that,

i. DEG = 1/2EDF

ii. EF =FG.


Answer :

Given: line DE || line GF


Ray EG and ray FG are bisectors of and respectively


To Prove: i.


ii.


Proof: Ray EG and ray FG are bisectors of and respectively.


So, DEG = GEF = 1/2 DEF ……………..(1)


DFG = GFM = 1/2 DFM ………..(2)


Also, EDF = DFG …..(3) [Alternate interior angles]


In ΔDEF


DFM = DEF + EDF


From (2) and (3)


2EDF = DEF + EDF


EDF = DEF


From (1)


EDF = 2DEG


DEG = 1/2 EDF


Hence, (i) is proved.


Line DE || line GF


From alternate interior angles


DEG = EGF …….(4)


From (1)


GEF = EGF


Since, in the ΔEGF sides opposite to equal angles are equal.


EF = FG


Hence, (ii) is proved.


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