Question

$\overrightarrow{ab}$ and $\overrightarrow{ac}$ are opposite rays. all of the following are true except ______.

$\bigcirc$ a, b, c are coplanar.

$\bigcirc$ ab = ac

$\bigcirc$ a is between b and c.

$\bigcirc$ a, b, c are collinear.

Explanation:

Step1: Recall opposite rays definition

Opposite rays have the same endpoint and form a straight line (collinear), so points \( A, B, C \) are collinear (and thus coplanar, as collinear points are coplanar). Also, \( A \) is between \( B \) and \( C \) for opposite rays.

Step2: Analyze \( AB = AC \)

Rays have infinite length, but \( AB \) and \( AC \) here refer to segments? Wait, no—opposite rays: \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) have endpoint \( A \), going in opposite directions. The segments \( AB \) and \( AC \) would be along the same line, but \( AB \) and \( AC \) as lengths—wait, no, actually, opposite rays mean \( B \) and \( C \) are on a straight line through \( A \), but \( AB \) and \( AC \) (as segments) would have \( A \) as a common endpoint, but \( AB \) and \( AC \) are not necessarily equal in length. Wait, no—wait, opposite rays: \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are opposite, so \( B \) and \( C \) are on a straight line with \( A \) between them? Wait, no: opposite rays have the same endpoint and their union is a line. So \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) have endpoint \( A \), and \( B \) and \( C \) are on a straight line, with \( A \) being the vertex, so \( A \) is not between \( B \) and \( C \)? Wait, no, wait: if \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are opposite rays, then \( B \) and \( C \) are on a straight line, and \( A \) is the common endpoint, so the line is \( BC \) with \( A \) on it? Wait, no, opposite rays: two rays with the same endpoint that form a straight line. So \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) have endpoint \( A \), and \( B \) and \( C \) are on a straight line, so \( A, B, C \) are collinear (so coplanar), and \( A \) is the vertex, so \( A \) is not between \( B \) and \( C \)? Wait, no, maybe I got that wrong. Wait, if \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are opposite rays, then they form a straight line, so \( B \) and \( C \) are on a straight line, and \( A \) is the point where the two rays meet, so \( A \) is between \( B \) and \( C \)? Wait, no, let's think: a ray \( \overrightarrow{AB} \) goes from \( A \) through \( B \), and \( \overrightarrow{AC} \) goes from \( A \) through \( C \). If they are opposite rays, then \( B \) and \( C \) are in opposite directions from \( A \), so the line is \( BC \) with \( A \) between \( B \) and \( C \)? Wait, no, \( A \) is the endpoint, so \( \overrightarrow{AB} \) is \( A \) to \( B \) (and beyond), \( \overrightarrow{AC} \) is \( A \) to \( C \) (and beyond). If they are opposite, then \( B \) and \( C \) are on a straight line, with \( A \) between them? Wait, no, that would mean \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are in opposite directions, so \( A \) is between \( B \) and \( C \). Wait, but then \( AB \) and \( AC \) are lengths from \( A \) to \( B \) and \( A \) to \( C \). But opposite rays don't imply \( AB = AC \). For example, \( A \) could be at the origin, \( B \) at \( (1,0) \), \( C \) at \( (-2,0) \): then \( \overrightarrow{AB} \) (along positive x-axis) and \( \overrightarrow{AC} \) (along negative x-axis) are opposite rays, but \( AB = 1 \), \( AC = 2 \), so \( AB eq AC \). So \( AB = AC \) is not necessarily true. The other options: collinear (yes, opposite rays form a line), coplanar (collinear points are coplanar), \( A \) is between \( B \) and \( C \) (yes, because opposite rays mean \( B \) and \( C \) are on a straight line with \( A \) in between? Wait, no, if \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are opposite, then \( B \) and \( C \) are on a straight line, and \( A \) is the common endpoint, so the line is \( BC \), and \( A \) is on \( BC \), so \( A \) is between \( B \) and \( C \). Wait, but in that case, \( AB \) and \( AC \) are segments from \( A \) to \( B \) and \( A \) to \( C \), but their lengths don't have to be equal. So the false statement is \( AB = AC \).

Answer:

\( AB = AC \) (the option with \( AB = AC \))