To solve the problem of proving that \( AC = BD \) given \( AB = CD \), we need to use the properties of equality and possibly the properties of segments in geometry. However, since this is a proof-based question and not a computational one, it is not suitable for a Python solution. Instead, we would typically write a formal proof using logical steps and geometric principles.
For a computational problem, please provide a different question.### Step 1: Understand the Given Information
We are given that \( AB = CD \) and need to prove that \( AC = BD \).
To prove \( AC = BD \), we need to understand the relationship between the segments \( AB \), \( CD \), \( AC \), and \( BD \).
Given:
\[ AB = CD \]
We need to show:
\[ AC = BD \]
Assume that points \( A \), \( B \), \( C \), and \( D \) are collinear in some geometric configuration. Without loss of generality, let's assume the points are arranged in a straight line such that \( A \) and \( C \) are endpoints of one segment, and \( B \) and \( D \) are endpoints of another segment.
If \( A \), \( B \), \( C \), and \( D \) are collinear, we can use the segment addition postulate. For example, if \( A \) and \( C \) are endpoints of one segment and \( B \) and \( D \) are endpoints of another segment, we can write:
\[ AC = AB + BC \]
\[ BD = BC + CD \]
Since \( AB = CD \), we can substitute \( CD \) for \( AB \) in the equation for \( BD \):
\[ BD = BC + AB \]
Now we have:
\[ AC = AB + BC \]
\[ BD = BC + AB \]
Since both equations are equal, we can conclude:
\[ AC = BD \]
Answered
