The angles of a quadrilateral are in AP whose common difference is ^@ 10^\circ ^@. Find the smallest angle of the quadrilateral.


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Answer:

^@ 75^\circ ^@

Step by Step Explanation:
  1. The angles of the quadrilateral are in AP with the common difference of ^@ 10^\circ ^@.

    Let the angles be ^@ x, x + 10^\circ, x + 20^\circ, ^@ and ^@ x + 30^\circ ^@.
  2. We know that the sum of all angles of a quadrilateral is ^@ 360^\circ ^@. Thus @^ \begin{aligned} & x + x + 10^\circ + x + 20^\circ + x + 30^\circ = 360^\circ \\ \implies & 4 x + 60^\circ = 360^\circ \\ \implies & 4 x = 360^\circ - 60^\circ \\ \implies & 4 x = 300^\circ \\ \implies & x = \dfrac { 300^\circ } { 4 } \\ \implies & x = 75^\circ \end{aligned} @^
  3. Let us now substitute the value of ^@ x ^@ to get the four angles. @^ \begin{aligned} & x = 75^\circ \\ & x + 10^\circ = 75^\circ + 10^\circ = 85^\circ \\ & x + 20^\circ = 75^\circ + 20^\circ = 95^\circ \\ & x + 30^\circ = 75^\circ + 30^\circ = 105^\circ \end{aligned} @^
  4. Hence, the smallest angle of the quadrilateral is ^@ 75^\circ ^@ .

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