To determine the four angles of an inscribed quadrilateral, we need to consider the given ratio of the arcs [tex]\(1: 2: 5: 4\)[/tex]. Let's break down the process step by step.
1. Sum of the Ratio Parts:
The sum of the parts in the ratio [tex]\(1: 2: 5: 4\)[/tex] is:
[tex]\[
1 + 2 + 5 + 4 = 12
\][/tex]
2. Total Degrees of Circle:
A circle has a total of [tex]\(360\)[/tex] degrees.
3. Calculating Each Angle:
Since the ratio indicates how the [tex]\(360\)[/tex] degrees are divided among the four angles, we multiply each part of the ratio by [tex]\(\frac{360}{12}\)[/tex] to find the corresponding angles.
- For the part '1':
[tex]\[
1 \times \frac{360}{12} = 30 \text{ degrees}
\][/tex]
- For the part '2':
[tex]\[
2 \times \frac{360}{12} = 60 \text{ degrees}
\][/tex]
- For the part '5':
[tex]\[
5 \times \frac{360}{12} = 150 \text{ degrees}
\][/tex]
- For the part '4':
[tex]\[
4 \times \frac{360}{12} = 120 \text{ degrees}
\][/tex]
So, the four angles of the inscribed quadrilateral are:
[tex]\[ 30 \text{ degrees}, 60 \text{ degrees}, 150 \text{ degrees}, \text{ and } 120 \text{ degrees.} \][/tex]
Therefore, you should fill the blanks in the question as follows:
The four angles of the quadrilateral are:
[tex]\[ 30 \,^\circ, 60 \,^\circ, 150 \,^\circ, \text{and } 120 \,^\circ. \][/tex]
