The sum of the measures of the interior angles of a convex n-gon is (n-2)*180^(∘).
In this case, expressions are given for the measures of the interior angles. We can write an equation where the sum of these expressions is equal to (n-2)180^(∘).
90^(∘)+4x+(8x-10^(∘)) +90^(∘)+(6x+4^(∘)) = (n-2)180^(∘)
Our polygon has 5 sides, so we can substitute 5 for n and solve our equation to find x.
Now, we need to solve the quadratic equation above. We will use the Quadratic Formula to do so.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
We first need to identify the values of a, b, and c.
x^2+17x -168=0 ⇕ 1x^2+ 17x+( - 168)=0
We see that a= 1, b= 17, and c= - 168. Let's substitute these values into the Quadratic Formula.
The solutions for this equation are x= -17± 312. Let's separate them into the positive and negative cases.
x=-17± 31/2
x_1=-17+31/2
x_2=-17-31/2
x_1=14/2
x_2=-48/2
x_1=7
x_2=-24
Using the Quadratic Formula, we found that the solutions of the given equation are x_1=7 and x_2=-24. Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, x = 7 is our solution.
c Consider the given diagram.
The diagram shows an isosceles triangle with one exterior angle of known angle. By the Base Angles Theorem we know that in isosceles triangle, angles opposite the congruent side have the same measure. Let's add this measure to the diagram.
Next, notice that the 134^(∘) and the last missing angle form a linear pair of angles. This means the sum of their measures equals 180^(∘). So, to find the missing angle, we can subtract 134^(∘) from 180^(∘).
180^(∘) - 134^(∘) = 46^(∘)
Let's add this angle to our diagram.
Recall that the angles in a triangle sum to 180^(∘). We can use this fact to write an equation for x. Let's do it!
( x+4^(∘))+( x+4^(∘)) + 46^(∘) = 180^(∘)
Finally, let's solve this equation for x.
d We want to find the value of x from the following diagram.
The diagram shows a parallelogram. In a parallelogram, the measures of adjacent angles sum to 180^(∘). With this, we can write the following equation.
(x+9^(∘))+(3x+15^(∘)) = 180^(∘)
Let's solve it for x!
