The sum of the measures of the interior angles of a convex n-gon is (n-2)*180^(∘).
We are given the measures of the other interior angles of the polygon, so let's write an equation where the sum of these measures is equal to (n-2)180.
2x + 4x-3^(∘) + 7x-6^(∘) + 3x+12^(∘) + x+10^(∘) = (n-2)180^(∘)
Our polygon has 5 sides, so we can substitute 5 for n. This means that the right-hand side of the equation will be equal to (5-2)180^(∘) = 540^(∘). Let's complete our equation with this information.
2x + 4x-3^(∘) + 7x-6^(∘) + 3x+12^(∘) + x+10^(∘) = 540^(∘)
Now that we have the equation for x, let's solve it!
Consider the diagram below. In order to find the value of x, we need to find the relationships between the angles and use the appropriate theorems and postulates.
It will be helpful to find the measure of one of the angles that makes a linear pair with the 4x+20^(∘) angle. Let's denote this measure by y.
Let's calculate each value one at a time. We will start with y.
Value of y
Looking at the diagram, we can see that the angles that measure y and 5x-2^(∘) are corresponding angles.
By the Corresponding Angles Postulate, corresponding angles formed by parallel lines and a transversal are congruent. From this we know that these two angles are congruent — their measures are the same.
y=5x-2^(∘)
The value of y is 5x-2^(∘) by the Corresponding Angles Theorem.
Value of x
Next, notice that the angles that measure 4x+20^(∘) and 5x-2^(∘) are supplementary angles.
Recall that by the Supplement Theorem, the sum of the measures of supplementary angles is 180^(∘).
4x+20^(∘)+ 5x-2^(∘)=180^(∘)
Let's solve this equation for x!
