In the given figure, AB||CD. Find the value of x. - Free Printable

Problem Analysis:

We are given a geometric figure with several intersecting lines and angles. The task is to find the value of the unknown angle \( x \) in degrees. Let's break down the problem step by step.

#### Given:
1. \( \angle BAC = 75^\circ \)
2. \( \angle FCE = 30^\circ \)
3. We need to find \( \angle AEC = x \).

#### Key Observations:
- The lines \( AB \) and \( DF \) intersect at point \( C \).
- The angles around point \( C \) are related through the properties of intersecting lines and the sum of angles on a straight line.

Step-by-Step Solution:

#### Step 1: Identify the relationship between \( \angle BAC \) and \( \angle DCA \)
The angles \( \angle BAC \) and \( \angle DCA \) are vertically opposite angles because they are formed by the intersection of lines \( AB \) and \( DF \) at point \( C \). Vertically opposite angles are always equal.

Thus:
\[
\angle DCA = \angle BAC = 75^\circ
\]

#### Step 2: Use the exterior angle property in \( \triangle CEF \)
In \( \triangle CEF \), \( \angle FCE \) is an exterior angle to \( \triangle CEF \) at vertex \( C \). The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles.

Here, \( \angle FCE \) is the exterior angle, and the two non-adjacent interior angles are \( \angle CEF \) and \( \angle EFC \). However, we are interested in finding \( \angle AEC \), which is the same as \( \angle CEF \) because they are the same angle.

Using the exterior angle theorem:
\[
\angle FCE = \angle CEF + \angle EFC
\]

We know:
\[
\angle FCE = 30^\circ
\]
\[
\angle EFC = \angle DCA = 75^\circ
\]

Substitute these values into the equation:
\[
30^\circ = \angle CEF + 75^\circ
\]

#### Step 3: Solve for \( \angle CEF \)
Rearrange the equation to solve for \( \angle CEF \):
\[
\angle CEF = 30^\circ - 75^\circ
\]
\[
\angle CEF = -45^\circ
\]

This result seems incorrect because an angle cannot be negative. Let's re-evaluate the setup. The correct approach is to recognize that \( \angle CEF \) is actually the difference between \( 180^\circ \) and the sum of the other two angles in the triangle.

#### Step 4: Correct Approach Using Straight Line Property
Since \( \angle FCE \) is an exterior angle, we can use the fact that the sum of angles on a straight line is \( 180^\circ \). The angle \( \angle AEC \) (or \( \angle CEF \)) is the supplement of the sum of the other two angles in the triangle.

Re-evaluate:
\[
\angle AEC = 180^\circ - (\angle FCE + \angle EFC)
\]
\[
\angle AEC = 180^\circ - (30^\circ + 75^\circ)
\]
\[
\angle AEC = 180^\circ - 105^\circ
\]
\[
\angle AEC = 75^\circ
\]

Final Answer:

\[
\boxed{45}
\]