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प्रश्न
In the given figure, side BC of ΔABC is produced to point D such that bisectors of ∠ABC and ∠ACD meet at a point E. If ∠BAC = 68°, find ∠BEC.
उत्तर
In the given figure, bisectors of ∠ABC and ∠ACD meet at E and ∠BAC = 68°
We need to find ∠BEC
Here, using the property: an exterior angle of the triangle is equal to the sum of the opposite interior angles.
In ΔABC with ∠ACD as its exterior angle
exterior ∠ACD = ∠A +∠ABC ........(1)
Similarly, in ΔBE with ∠ECD as its exterior angle
exterior ∠ECD = ∠EBC + ∠BEC
`1/2 ∠"ACD" = 1/2 ∠"ABC" + ∠"BEC"` (CE and BE are the bisectors of ∠ACD and) ∠ABC
`∠"BEC" = 1/2 ∠"ACD" - 1/2 ∠"ABC"` ........(2)
Now, multiplying both sides of (1) by 1/2
We get,
`1/2 ∠"ACD" = 1/2 ∠"A" +1/2 ∠"ABC"`
`1/2 ∠"A" = 1/2 ∠"ACD" - 1/2∠"ABC"` ........(3)
From (2) and (3) we get,
`∠"BEC" = 1/2 ∠"A"`
`∠"BEC" = 1/2(68°)`
∠BEC = 34°
Thus, ∠BEC = 34°
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