Answer:
∠DCB = 128° and ∠ECD = 52°
Step-by-step explanation:
∠DCB and ∠ACE are vertically opposite angles (or vertical angles), meaning they are created by the same two intersecting lines and thus share the same angle measurement. This means the measurement of ∠DCB is also 3x + 5.
∠DCB and ∠ACB are supplementary angles meaning they are angles that share the same line dissected by a transverse angle. As such, the sum of their angles is 180°. Above, we established ∠DCB = 3x + 5. Given in the diagram, ∠ACB = x + 11. We can thus, created an equation, combine like terms, solve for x, and then solve for the angle.
x + 11 + 3x + 5 = 180
4x + 16 = 180
4x = 180 - 16
4x = 164
4x / 4 = 164 / 4
x = 41
Now we plug this value for x into the equation for ∠DCB
∠DCB = 3x + 5
3(41) + 5
123 + 5
∠DCB = 128°
∠ECD and ∠ACB are also vertically opposite angles so because they share the ∠ACB's measurement equation, x + 11, we can use the value of x to solve for ∠ECD's exact measurement.
∠ECD = x + 11
(41) + 11
∠ECD = 52°