https://file.aiquickdraw.com/mathbot/file/1726827821915g2y4wht4.jpg

In this problem, we are given that \( m\angle ACD = 40^\circ \), and we need to determine \( m\angle DCB \).

From the image, it's clear that points \( A \), \( C \), and \( B \) are collinear, forming a straight line. Therefore, the angles \( \angle ACD \) and \( \angle DCB \) form a linear pair. A linear pair of angles always sums up to \( 180^\circ \).

Since \( m\angle ACD = 40^\circ \), the measure of \( m\angle DCB \) can be found using the following equation:

\[
m\angle ACD + m\angle DCB = 180^\circ
\]

Substituting the known value:

\[
40^\circ + m\angle DCB = 180^\circ
\]

Solving for \( m\angle DCB \):

\[
m\angle DCB = 180^\circ - 40^\circ = 140^\circ
\]

Thus, \( m\angle DCB = 140^\circ \).

### Do you want more details or have any further questions?

Here are some related questions for further practice:

1. What is the measure of the third angle in a triangle if two angles are \( 40^\circ \) and \( 80^\circ \)?
2. If two angles form a linear pair and one of them is \( 65^\circ \), what is the measure of the other angle?
3. What is the sum of the interior angles of a quadrilateral?
4. How can you determine if two angles are supplementary?
5. In a straight angle, if one of the angles is \( 75^\circ \), what is the measure of the other?

**Tip:** When two angles form a straight line (linear pair), their measures always add up to \( 180^\circ \).

This problem explains how to determine the measure of angle DCB when angle ACD is given as 40° and both angles form a linear pair. The solution includes key geometry concepts such as linear pair of angles and angle relationships, suitable for students in Grades 6-8.

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