The angles of a quadrilateral are in the ratio \[3:4:5:6\]. How do you find the angles of the quadrilateral?

Answer
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Hint: Given, the four angles of a quadrilateral are in the ratio \[3:4:5:6\]. We will assume the four angles to be \[3x\], \[4x\], \[5x\] and \[6x\] respectively. Using the angle sum property of a quadrilateral we form an equation in terms of \[x\]. We will solve the equation to find the value of \[x\] and we will use the value of \[x\] to find the measure of the four angles of the quadrilateral.

Complete step-by-step answer:

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It is given that the angles of the quadrilateral are in the ratio \[3:4:5:6\].
Let the four angles of the given quadrilateral be \[3x\], \[4x\], \[5x\] and \[6x\] respectively.
Now, from the angle sum property of a quadrilateral we know that the sum of the measures of the four interior angles of a quadrilateral is always \[{360^ \circ }\].
Using this, we can write
\[ \Rightarrow 3x + 4x + 5x + 6x = {360^ \circ }\]
Adding the like terms in the above equation, we get
\[ \Rightarrow 18x = {360^ \circ }\]
Dividing both the sides of the equation by \[18\], we get
\[ \Rightarrow \dfrac{{18x}}{{18}} = \dfrac{{{{360}^ \circ }}}{{18}}\]
Cancelling the common terms from the numerator and the denominator, we get
\[ \Rightarrow x = {20^ \circ }\]
Finally, we will substitute the value of \[x\] to find the measures of all the angles of the quadrilateral.
On doing this, we get the four angles of the given quadrilateral as \[\left( {3 \times {{20}^ \circ }} \right)\], \[\left( {4 \times {{20}^ \circ }} \right)\], \[\left( {5 \times {{20}^ \circ }} \right)\] and \[\left( {6 \times {{20}^ \circ }} \right)\] i.e., \[{60^ \circ }\], \[{80^ \circ }\], \[{100^ \circ }\] and \[{120^ \circ }\] respectively.
Therefore, the angles of the quadrilateral are \[{60^ \circ }\], \[{80^ \circ }\], \[{100^ \circ }\] and \[{120^ \circ }\].

Note: Here, the question is of a quadrilateral. Like the sum of the measure of the angles of a quadrilateral is always \[{360^ \circ }\]. Similarly, the sum of the measure of the angles of a triangle having three sides is always \[{180^ \circ }\], the sum of the measure of the angles of a pentagon having five sides is always \[{540^ \circ }\] and the sum of the measure of the angles of a hexagon having six sides is always \[{720^ \circ }\].