The problem describes a triangle $ABC$ and asks how the angles of the triangle change when point $C$ is moved along the line segment $BC$. Specifically, we need to determine how $m\angle ABC$, $m\angle BAC$, and $m\angle ACB$ change when $C$ moves toward $B$ and when $C$ moves away from $B$.

1. Problem Description

The problem describes a triangle $ABC$ and asks how the angles of the triangle change when point $C$ is moved along the line segment $BC$. Specifically, we need to determine how $m\angle ABC$, $m\angle BAC$, and $m\angle ACB$ change when $C$ moves toward $B$ and when $C$ moves away from $B$.

2. Solution Steps

When point $C$ moves toward point $B$ along $BC$:

* $m\angle ABC$: Since $C$ is approaching $B$, the length of $BC$ decreases. The angle $ABC$ will decrease as $C$ approaches $B$.

* $m\angle BAC$: As $C$ approaches $B$, the angle $BAC$ increases, since the length $AC$ is decreasing, thereby widening the angle at A.

* $m\angle ACB$: Since the sum of angles in a triangle is 180 degrees, $m\angle ACB$ = $180^\circ - m\angle ABC - m\angle BAC$. Since $m\angle ABC$ decreases and $m\angle BAC$ increases as $C$ approaches $B$, but $m\angle ABC$ is approaching $0$, and $m\angle BAC$ is approaching $180-m\angle BAC$, therefore $m\angle ACB$ will approach 0 as $C$ approaches $B$.

When point $C$ moves away from point $B$ along $BC$:

* $m\angle ABC$: As $C$ moves away from $B$, the length of $BC$ increases. Therefore $m\angle ABC$ increases.

* $m\angle BAC$: As $C$ moves away from $B$, the angle $BAC$ decreases.

* $m\angle ACB$: As $C$ moves away from $B$, $m\angle ABC$ increases and $m\angle BAC$ decreases. Therefore, $m\angle ACB$ increases.

3. Final Answer

If you move point $C$ toward point $B$ along $BC$, $m\angle ABC$ decreases, $m\angle BAC$ increases, and $m\angle ACB$ decreases.

If you move point $C$ away from point $B$ along $BC$, $m\angle ABC$ increases, $m\angle BAC$ decreases, and $m\angle ACB$ increases.