Given that the four vertices of an inscribed quadrilateral divide a circle in the ratio [tex]\( 1:2:5:4 \)[/tex]:
### Step-by-Step Solution
1. Understanding the total angles:
- An inscribed quadrilateral is a quadrilateral where all vertices lie on a circle, thus making it a cyclic quadrilateral.
- The sum of angles in any quadrilateral is [tex]\(360^\circ\)[/tex].
2. Distribution of the total angle:
- The circle is divided into parts in the ratio [tex]\(1:2:5:4\)[/tex].
- The total number of parts in this ratio is [tex]\(1 + 2 + 5 + 4 = 12\)[/tex].
3. Calculating each angle:
- Each part represents a fraction of the total [tex]\(360^\circ\)[/tex].
[tex]\[
\text{First angle: } \frac{1}{12} \times 360^\circ = 30^\circ
\][/tex]
[tex]\[
\text{Second angle: } \frac{2}{12} \times 360^\circ = 60^\circ
\][/tex]
[tex]\[
\text{Third angle: } \frac{5}{12} \times 360^\circ = 150^\circ
\][/tex]
[tex]\[
\text{Fourth angle: } \frac{4}{12} \times 360^\circ = 120^\circ
\][/tex]
### Result:
The four angles of the inscribed quadrilateral are [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], [tex]\(150^\circ\)[/tex], and [tex]\(120^\circ\)[/tex].
So, select the correct answers as:
- The four angles of the quadrilateral are `30 ∘`, `60 ∘`, `150 ∘`, and `120 ∘ `.
