Answer :
Sure, let's find the value of the smallest angle of the hexagon given the ratio of the internal angles is 2:3:4:4:5:6.
### Step 1: Find the Sum of Internal Angles of the Hexagon
The sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula:
[tex]\[ \text{Sum of internal angles} = (n - 2) \times 180^\circ \][/tex]
For a hexagon ([tex]\( n = 6 \)[/tex]):
[tex]\[ \text{Sum of internal angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \][/tex]
### Step 2: Understand the Ratio of the Angles
The ratio of the angles is given as 2:3:4:4:5:6. To simplify working with these ratios, let's sum them up.
So, [tex]\( 2 + 3 + 4 + 4 + 5 + 6 = 24 \)[/tex]
### Step 3: Determine the Value of One Part in the Ratio
Now, we know that the total sum of the angles (720 degrees) corresponds to the sum of the ratios (24 parts). Thus, the value of one part is:
[tex]\[ \text{Value of one part} = \frac{720^\circ}{24} = 30^\circ \][/tex]
### Step 4: Calculate the Smallest Angle
The smallest ratio given is 2. Therefore, to find the smallest angle:
[tex]\[ \text{Smallest angle} = 2 \times (\text{Value of one part}) = 2 \times 30^\circ = 60^\circ \][/tex]
### Conclusion
The smallest angle in the hexagon, given the ratio 2:3:4:4:5:6, is [tex]\( 60^\circ \)[/tex].
### Step 1: Find the Sum of Internal Angles of the Hexagon
The sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula:
[tex]\[ \text{Sum of internal angles} = (n - 2) \times 180^\circ \][/tex]
For a hexagon ([tex]\( n = 6 \)[/tex]):
[tex]\[ \text{Sum of internal angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \][/tex]
### Step 2: Understand the Ratio of the Angles
The ratio of the angles is given as 2:3:4:4:5:6. To simplify working with these ratios, let's sum them up.
So, [tex]\( 2 + 3 + 4 + 4 + 5 + 6 = 24 \)[/tex]
### Step 3: Determine the Value of One Part in the Ratio
Now, we know that the total sum of the angles (720 degrees) corresponds to the sum of the ratios (24 parts). Thus, the value of one part is:
[tex]\[ \text{Value of one part} = \frac{720^\circ}{24} = 30^\circ \][/tex]
### Step 4: Calculate the Smallest Angle
The smallest ratio given is 2. Therefore, to find the smallest angle:
[tex]\[ \text{Smallest angle} = 2 \times (\text{Value of one part}) = 2 \times 30^\circ = 60^\circ \][/tex]
### Conclusion
The smallest angle in the hexagon, given the ratio 2:3:4:4:5:6, is [tex]\( 60^\circ \)[/tex].
