Question

The angles of a quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the quadrilateral.

Answer 1

Hint: Here, we need to find the measure of four angles of the quadrilateral. We will assume the four angles to be \[3x\] , \[5x\], \[9x\], and \[13x\] respectively. We will apply the angle sum property of a quadrilateral to form an equation in terms of \[x\]. Then, we will solve the equation to find the value of \[x\]. Then we will use the value of \[x\] to find the measure of all the angles of the quadrilateral.

Complete step-by-step answer:
We will use the angle sum property of a quadrilateral to find the measure of all the angles of the quadrilateral.
It is given that the angles of the quadrilateral are in the ratio 3: 5: 9: 13.
Let the four angles be \[3x\], \[5x\], \[9x\], and \[13x\] respectively.
Now, the angle sum property of a quadrilateral states that the sum of the measures of the four interior angles of a triangle is always \[360^\circ \].
Thus, the sum of the two four angles will be equal to \[360^\circ \].
Therefore, we get
\[3x + 5x + 9x + 13x = 360^\circ \]
We will solve this equation to find the value of \[x\].
Adding the like terms in the equation, we get
\[ \Rightarrow 30x = 360^\circ \]
Dividing both sides of the equation by 30, we get
\[\Rightarrow \dfrac{{30x}}{{30}} = \dfrac{{360^\circ }}{{30}} \\
   \Rightarrow x = 12^\circ \\\]
\[\therefore \] We get the value of \[x\] as \[12^\circ \].
Finally, we will substitute the value of \[x\] to find the measures of all the angles of the quadrilateral.
Substituting \[x = 12^\circ \] in \[3x\], we get the first angle of the quadrilateral as
\[3x = 3 \times 12^\circ = 36^\circ \]
Substituting \[x = 12^\circ \] in \[5x\], we get the second angle of the quadrilateral as
\[5x = 5 \times 12^\circ = 60^\circ \]
Substituting \[x = 12^\circ \] in \[9x\], we get the third angle of the quadrilateral as
\[9x = 9 \times 12^\circ = 108^\circ \]
Substituting \[x = 12^\circ \] in \[13x\], we get the fourth angle of the quadrilateral as
\[13x = 13 \times 12^\circ = 156^\circ \]
\[\therefore \] The measures of all the angles of the quadrilateral are \[36^\circ \], \[60^\circ \], \[108^\circ \], and \[156^\circ \].

Note: We have subtracted and divided both sides of an equation by the same number. This is because equals added to, subtracted from, multiplied by, or divided by equals, remains equal.
We can verify our answer using the information given in the question.
The ratio of the four angles of the quadrilateral is 36: 60: 108: 156.
Dividing by 12, we can simplify the ratio as 3: 5: 9: 13.
This is the ratio given in the question.
Hence, we have verified the answer.