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### Given:
- Lines \( AB \parallel CD \).
- \( \angle ABC = 35^\circ \) and \( \angle BAC = 115^\circ \).

We need to find:
1. \( m\angle ACE \)
2. \( m\angle DCB \)
3. \( m\angle ACB \)

### Solution:

#### a. What is \( m\angle ACE \)?
From the diagram, we can see that \( AB \parallel CD \), and line \( AC \) acts as a transversal. Therefore, by the Alternate Interior Angles Theorem:
\[
m\angle ACE = m\angle BAC = 115^\circ.
\]
Thus, \( m\angle ACE = 115^\circ \).

#### b. What is \( m\angle DCB \)?
Since lines \( AB \parallel CD \) and \( BC \) is the transversal, \( \angle ABC \) and \( \angle DCB \) are alternate interior angles. Therefore:
\[
m\angle DCB = m\angle ABC = 35^\circ.
\]
Thus, \( m\angle DCB = 35^\circ \).

#### c. What is \( m\angle ACB \)?
We know that the sum of the angles in any triangle is \( 180^\circ \). In triangle \( ACB \), the three angles are \( \angle BAC \), \( \angle ABC \), and \( \angle ACB \). Therefore, we can find \( \angle ACB \) by subtracting the other two from 180:
\[
m\angle ACB = 180^\circ - (m\angle ABC + m\angle BAC) = 180^\circ - (35^\circ + 115^\circ) = 30^\circ.
\]
Thus, \( m\angle ACB = 30^\circ \).

### Final Answers:
1. \( m\angle ACE = 115^\circ \)
2. \( m\angle DCB = 35^\circ \)
3. \( m\angle ACB = 30^\circ \)

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the relationship between corresponding angles when two lines are parallel?
2. How do you use the Alternate Interior Angles Theorem to solve parallel line problems?
3. How would the angles change if \( AB \) and \( CD \) were not parallel?
4. Can you explain the importance of the transversal in determining angle relationships?
5. How would you find the angles in a non-right triangle with different side lengths?

### Tip:
Always remember that the sum of the interior angles of any triangle is always 180°, a useful fact in solving for unknown angles.

In the diagram, lines AB and CD are parallel. Angle ABC measures 35° and angle BAC measures 115°. What is m∠ACE, m∠DCB, and m∠ACB?

Learn how to solve for angles in a triangle where lines AB and CD are parallel, with angles ABC = 35° and BAC = 115°. This problem demonstrates the use of the Alternate Interior Angles Theorem and the sum of angles in a triangle. Ideal for Grade 8 students.

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