A) \[15{}^\circ and 70{}^\circ \]
B) \[10{}^\circ and 160{}^\circ \]
C) \[20{}^\circ and 130{}^\circ \]
D) \[20{}^\circ and 125{}^\circ \]
E) None of these
Correct Answer: C
Solution :
Explanation In \[\Delta ABC, \angle A+\angle B+\angle C=180{}^\circ \] \[80{}^\circ + 60{}^\circ + 2x{}^\circ = 180{}^\circ \] \[\Rightarrow \,\,\,\,\,2x{}^\circ =40{}^\circ \] \[\operatorname{x}{}^\circ = 20{}^\circ \] In ZBDC, \[\angle DBC + \angle DCB + \angle BDC = 180{}^\circ \] \[\Rightarrow \,\,\,\frac{1}{2}\,\angle ABC+\,\,\frac{1}{2}\angle ACB+\angle BDC=180{}^\circ \] (BD and CD bisect \[\angle B\text{ }and\text{ }\angle D\]) \[\Rightarrow \,\,\,\,\frac{1}{2}\left( \angle ABC + \angle ACB \right) + \angle BDC = 180{}^\circ \] \[\Rightarrow \,\,\,\frac{1}{2}\left( 60{}^\circ + 2x{}^\circ \right) + y{}^\circ = 180{}^\circ \] \[\Rightarrow \,\,\,\frac{1}{2}\left( 100{}^\circ \right) + y{}^\circ = 180{}^\circ \] \[\Rightarrow \,\,\,\,\operatorname{y}{}^\circ = 180{}^\circ - 50{}^\circ = 130{}^\circ .\]
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