Sure, let's check all the given equations step by step to see which ones are equivalent:
Given equations:
1. [tex]\( a = 180(n - 2) \)[/tex]
2. [tex]\( n = \frac{a}{180} + 1 \)[/tex]
3. [tex]\( n = \frac{a}{180} + 2 \)[/tex]
4. [tex]\( n = \frac{a + 360}{180} \)[/tex]
### Step 1: Simplify Equation 1
[tex]\[ a = 180(n - 2) \][/tex]
We solve for [tex]\( n \)[/tex]:
[tex]\[ a = 180n - 360 \][/tex]
[tex]\[ 180n = a + 360 \][/tex]
[tex]\[ n = \frac{a + 360}{180} \][/tex]
So, we have:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
### Step 2: Compare the Result with Other Equations
#### Compare with Equation 2:
Equation 2 is:
[tex]\[ n = \frac{a}{180} + 1 \][/tex]
Clearly, the right-hand side [tex]\(\frac{a}{180} + 1\)[/tex] is not equal to [tex]\(\frac{a + 360}{180}\)[/tex]. Therefore, Equation 2 is not equivalent to Equation 1.
#### Compare with Equation 3:
Equation 3 is:
[tex]\[ n = \frac{a}{180} + 2 \][/tex]
Let's test this result:
[tex]\[ \frac{a + 360}{180} = \frac{a}{180} + \frac{360}{180} = \frac{a}{180} + 2 \][/tex]
Clearly, [tex]\(\frac{a + 360}{180}\)[/tex] is equal to [tex]\(\frac{a}{180} + 2\)[/tex]. Therefore, Equation 3 is equivalent to Equation 1.
#### Compare with Equation 4:
Equation 4 is:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
We already obtained [tex]\( n = \frac{a + 360}{180} \)[/tex] from Equation 1. Therefore, Equation 4 is equivalent to Equation 1.
### Summary
After comparing, we find that Equations 3 and 4 are equivalent to Equation 1. Therefore, the equivalent equations are:
1. [tex]\( a = 180(n - 2) \)[/tex]
2. [tex]\( n = \frac{a}{180} + 2 \)[/tex]
3. [tex]\( n = \frac{a + 360}{180} \)[/tex]
Equation 2, [tex]\( n = \frac{a}{180} + 1 \)[/tex], is not equivalent to the others.
