To solve this problem, we need to determine the angles formed at the vertices of an inscribed quadrilateral given that they divide the circle in the ratio [tex]\(1:2:5:4\)[/tex].
1. The total sum of all parts in the ratio can be calculated as:
[tex]\[
1 + 2 + 5 + 4 = 12
\][/tex]
2. A full circle has [tex]\(360^\circ\)[/tex]. To find the angle corresponding to each part of the ratio, we divide the circle's total degrees by the total sum of the parts:
[tex]\[
\frac{360^\circ}{12} = 30^\circ
\][/tex]
3. Now, multiply each of the parts in the ratio by [tex]\(30^\circ\)[/tex] to find the angles:
- For the ratio 1 part:
[tex]\[
1 \times 30^\circ = 30^\circ
\][/tex]
- For the ratio 2 parts:
[tex]\[
2 \times 30^\circ = 60^\circ
\][/tex]
- For the ratio 5 parts:
[tex]\[
5 \times 30^\circ = 150^\circ
\][/tex]
- For the ratio 4 parts:
[tex]\[
4 \times 30^\circ = 120^\circ
\][/tex]
Therefore, the four angles of the quadrilateral are [tex]\(30^\circ, 60^\circ, 150^\circ\)[/tex], and [tex]\(120^\circ\)[/tex].
Thus, the completed statement is:
The four angles of the quadrilateral are [tex]\(30^\circ, 60^\circ, 150^\circ\)[/tex], and [tex]\(120^\circ\)[/tex].
