Final answer:
To find the measure of angle BEC, we can set up an equation and solve for x. The value of x can help us find the measures of angles AEC and DEB, and then we can subtract the measure of angle DEB from 180 degrees to find the measure of angle BEC.
Step-by-step explanation:
To find the measure of angle BEC, we need to first find the value of x. Since angle AEC and angle DEB intersect at point E, the sum of their measures should be equal to 180 degrees. So, we can set up the equation:
7x + 10 + x^2 + 4 = 180
Simplifying the equation, we get:
x^2 + 7x + 14 = 180
Next, we subtract 180 from both sides:
x^2 + 7x - 166 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Once we find the value of x, we can substitute it back into the expressions for angle AEC and angle DEB to find their measures. Finally, to find the measure of angle BEC, which is opposite to angle DEB, we subtract the measure of angle DEB from 180 degrees.
Once you've found the value of x, you can calculate the respective expressions and deduce the measure of angle BEC. The answer choices provided are 104°, 115°, 120°, and 128°, so make sure to verify the calculated result with these options.