Final answer:
By using the properties of angles in a rectangle, we derived an equation to find the value of x and determined that x is 14. Then we deduced that angle BCA is equal to 44 degrees, as it is the opposite angle to ABD in rectangle ABCD.
Step-by-step explanation:
We are given the angles ABD and CBD in a rectangle ABCD, and need to find the value of x and angle BCA. Since ABCD is a rectangle, angle ABD plus angle CBD equals 90 degrees (as they are complementary angles on the diagonal BD). We can then set up the equation:
ABD (3x + 2) + CBD (2x + 18) = 90
By combining like terms, we get:
5x + 20 = 90
Subtracting 20 from both sides:
5x = 70
Dividing by 5:
x = 14
Now that we have the value of x, we can find the angle ABD:
ABD = 3(14) + 2 = 42 + 2 = 44 degrees.
Since ABCD is a rectangle, angle BCA is also equal to angle ABD, as opposite angles in a rectangle are equal. Therefore, the angle BCA is also 44 degrees.