To solve this problem, we need to determine the angles of a quadrilateral inscribed in a circle whose vertices divide the circle in the ratio [tex]\(1: 2: 5: 4\)[/tex]. Here's the step-by-step solution:
1. The sum of the angles of any quadrilateral is [tex]\(360\)[/tex] degrees. This is a key fact to remember for our calculations.
2. The ratio of the division of the circle by the vertices of the inscribed quadrilateral is given as [tex]\(1: 2: 5: 4\)[/tex].
3. To find each angle of the quadrilateral, we will first sum up the parts of the ratio:
[tex]\[
1 + 2 + 5 + 4 = 12
\][/tex]
4. Each part of the ratio represents a portion of the total [tex]\(360\)[/tex] degrees of the circle, proportional to its share in the ratio.
5. Now, we determine the actual angle corresponding to each part of the ratio:
[tex]\[
\text{First angle} = \frac{1}{12} \times 360 = 30 \text{ degrees}
\][/tex]
[tex]\[
\text{Second angle} = \frac{2}{12} \times 360 = 60 \text{ degrees}
\][/tex]
[tex]\[
\text{Third angle} = \frac{5}{12} \times 360 = 150 \text{ degrees}
\][/tex]
[tex]\[
\text{Fourth angle} = \frac{4}{12} \times 360 = 120 \text{ degrees}
\][/tex]
Thus, the four angles of the inscribed quadrilateral are [tex]\(30\)[/tex] degrees, [tex]\(60\)[/tex] degrees, [tex]\(120\)[/tex] degrees, and [tex]\(150\)[/tex] degrees.
So, filling the result into the provided format, we get:
The four angles of the quadrilateral are
[tex]\(30 \degree\)[/tex] (because the ratio is 1 part),
[tex]\(60 \degree\)[/tex] (because the ratio is 2 parts),
[tex]\(150 \degree\)[/tex] (because the ratio is 5 parts), and
[tex]\(120 \degree\)[/tex] (because the ratio is 4 parts).
