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In this diagram, we are given that \( AB \) and \( CD \) are straight lines, and the angles at point \( O \) are provided, with two \( 30^\circ \) angles marked. We need to find the value of \( x \) (the unknown angle).

### Solution:

1. Since \( AB \) and \( CD \) are straight lines, we know that the angles around point \( O \) must add up to \( 360^\circ \).

2. We are given:
   - Two \( 30^\circ \) angles (let’s label these as \( \angle COE \) and \( \angle EOF \)).
   - The unknown angle \( x \).
   - An angle \( y \) (unlabeled but adjacent to the other angles).

3. Let’s denote the angles in the following way around point \( O \):
   \[
   \angle AOD = y, \quad \angle COE = 30^\circ, \quad \angle EOF = 30^\circ, \quad \angle FOD = x
   \]

4. Since the angles at a point add up to \( 360^\circ \), we can write the equation:
   \[
   y + 30^\circ + x + 30^\circ = 180^\circ
   \]


If shown, AB and CD are straight lines. Find x in each case.

Explore how to find unknown angles in a geometry problem involving straight lines and a central point. Understand the application of the sum of angles around a point, with a focus on angle properties and relationships. Suitable for students in Grades 8-10.

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