Question: Smallest angle of a triangle is equal to two-third the smallest angle of a quadrilateral. The ratio between the angles of the quadrilateral is 3: 4: 5: 6. Largest angle of the triangle is twice its smallest angle. What is the sum of second largest angle of the triangle and largest angle of the quadrilateral?
Options:
A) $ 160{}^\circ $
B) $ 180{}^\circ $
C) $ 190{}^\circ $
D) $ 170{}^\circ $
E) None of these
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Answer:
Correct Answer: B
Solution:
- [b] Let the angles of the quadrilateral be and $ 3x, $ $ 4x, $ $ 5x $ and $ 6x $ respectively.
Then, $ 3x+4x+5x+6x=360{}^\circ $
$ \Rightarrow $ $ 18x=360{}^\circ $
$ \Rightarrow $ $ x=20{}^\circ $
$ \therefore $ Smallest angle of the triangle
$ =3\times 20\times \frac{2}{3}=40{}^\circ $
$ \therefore $ Largest angle of the triangle $ =40{}^\circ \times 2=80{}^\circ $
$ \therefore $ Second largest angle of triangle
$ =180{}^\circ -(40{}^\circ +80{}^\circ )=60{}^\circ $
and largest angle of the quadrilateral
$ =6x=6\times 20{}^\circ $
$ =120{}^\circ $
Hence, required sum $ =60{}^\circ +120{}^\circ =180{}^\circ $
BETA
