Final answer:
To solve for x and find each angle measure, we can apply the angle sum and exterior angle theorems. We set up an equation and solve for x using algebraic techniques. Substituting the value of x back into the expressions gives us the angle measures. The angle measures are approximately 105.428 degrees, 70.429 degrees, 24.428 degrees, and 105.428 degrees respectively.
Step-by-step explanation:
To solve for x and find each angle measure, we can apply the angle sum and exterior angle theorems. Let's start by setting up our equations. The angle sum theorem states that the sum of the angles in a triangle is 180 degrees. The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Given the expressions (18x - 15), (13x - 11), and (4x + 1), we can set up the equation: (18x - 15) + (13x - 11) + (4x + 1) = 180. Now, we can solve for x:
18x - 15 + 13x - 11 + 4x + 1 = 180
35x - 25 = 180
35x = 205
x = 205/35
x = 5.857
Now, we can substitute the value of x back into the expressions to find the angle measures.
m∠CAB = 18x - 15 = 18(5.857) - 15 = 105.428
m∠ABC = 13x - 11 = 13(5.857) - 11 = 70.429
m∠ACB = 4x + 1 = 4(5.857) + 1 = 24.428
m∠DCB = m∠CAB = 105.428