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To solve this problem, let’s break down the key information provided:

### Given:
1. \( AB \parallel EC \) (AB is parallel to EC).
2. \( \angle DAE = 70^\circ \) and \( \angle CED = 30^\circ \).
3. \( AD \) and \( CE \) intersect at point \( F \).
4. \( \angle BCD = \theta \).
5. We need to express angles \( m \) (which is \( \angle AFE \)) and \( k \) (which is \( \angle CFE \)) in terms of \( \theta \).

### Step-by-step solution:

#### Step 1: Understanding the parallel lines \( AB \parallel EC \)
Since \( AB \parallel EC \), we can apply alternate interior angles.

- \( \angle ABC = \angle BCD = \theta \) (since they are alternate interior angles due to the parallel lines and transversal \( BC \)).

#### Step 2: Analyzing angle relations in the circle
- The angle \( \angle CED = 30^\circ \), which is given.
- The angle \( \angle DAE = 70^\circ \), which is also given.

Since \( A, B, C, D, E \) lie on the circle, we can apply properties of angles in a cyclic quadrilateral.

#### Step 3: Cyclic Quadrilateral Property
In a cyclic quadrilateral, opposite angles add up to \( 180^\circ \).

- So, \( \angle BAE + \angle BCD = 180^\circ \).

Given that \( \angle BCD = \theta \), we have:
\[
\angle BAE = 180^\circ - \theta
\]

But \( \angle BAE = \angle DAE = 70^\circ \) (since \( AB \parallel EC \)).

Thus, \( 180^\circ - \theta = 70^\circ \), which gives:
\[
\theta = 110^\circ
\]

#### Step 4: Finding \( m \) and \( k \)
Now that \( \theta = 110^\circ \), we can express angles \( m \) and \( k \).

- \( m = \angle AFE \) is the angle subtended by arc \( AE \) at point \( F \). From the circle theorem (angle at the center is twice the angle at the circumference), \( m = \frac{1}{2} \times 70^\circ = 35^\circ \).

- \( k = \angle CFE \) is the angle subtended by arc \( CE \). Using the same theorem, \( k = \frac{1}{2} \times 30^\circ = 15^\circ \).

### Final Answer:
- \( m = 35^\circ \)
- \( k = 15^\circ \)

Do you need further details or clarifications?

### Related Questions:
1. What are the properties of a cyclic quadrilateral?
2. How do parallel lines affect angle relationships in a circle?
3. What is the relationship between central and inscribed angles in a circle?
4. How can the alternate interior angle theorem be applied in circles?
5. What are other angle properties related to intersecting chords in a circle?

### Tip:
Always look for cyclic quadrilateral properties when dealing with circles, as opposite angles add to \(180^\circ\).

In the figure, ABCDE is a circle. It is given that AB // EC, ∠DAE = 70° and ∠CED = 30°. AD and CE intersect at point F. Let ∠BCD be θ. Express m and k in terms of θ.

This problem explores the relationships between angles in a circle with parallel lines and a cyclic quadrilateral. Learn how to express angles m and k in terms of θ using circle theorems and properties of cyclic quadrilaterals.

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