If two lines are parallel to the same line, then they are parallel to each other. Is there a specific name for this theorem when being used in proofs?

For a line:

y = a*x + b

where a is the slope, and b is the y-intercept.

Any other line with the same slope but different y-intercept will be parallel to this line.

Then if we have two lines like:

y1 = a*x + c

and

y2 = a*x + k

Both of them are parallel to our first line, and as you can see, both of them have the same slope and different y-intercept, then these lines are also parallel.

Now, the statement says that:

"If two lines are parallel to the same line..."

Then we can have the lines:

y1 = a*x + k

y2 = a*x + k

Both of them are parallel to our first line (same slope)

Buth these lines are the same line, then these lines are not parallel to each other (a line is not parallel to itself, because it intersects itself in infinity points), then the actual statement should be:

"If two different lines are parallel to the same line, then they are parallel to each other".

Now, this theorem does not have a particular name, just call it as the above statement.

10 glasses of juice could be made with 5 carrots, 1,500 orange juice, 2,000 papaya juice.

whats the question?

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