Question

three points a, b, and c lie on the same line. read the statements. a: if point b is between points a and c, then $overline{ab}+overline{bc}=overline{ac}$. b: if $overline{ab}+overline{bc}=overline{ac}$, then point b is between points a and c. c: if $overline{ab}+overline{bc}
eqoverline{ac}$, then point b is not between points a and c. select all the true statements. statement b is the inverse of statement a. statement c is the inverse of statement a. statement b is the converse of statement a. statement c is the converse of statement a. statement b is the contrapositive of statement a. statement c is the contrapositive of statement a.

Explanation:

1. Recall the definitions of inverse, converse, and contra - positive in logic applied to geometric statements.

• The inverse of a conditional statement "if p then q" is "if not p then not q".
• The converse is "if q then p".
• The contra - positive is "if not q then not p".
• For statement A: "If point B is between points A and C, then $\overline{AB}+\overline{BC}=\overline{AC}$".
• The inverse (Statement B) is "If $\overline{AB}+\overline{BC}=\overline{AC}$, then point B is between points A and C". This is false because the points could be collinear in a non - betweenness order (e.g., A, C, B) and still satisfy the length equation if we consider directed lengths.
• The converse of A is the same as the inverse here.
• The contra - positive (Statement C) of A: "If $\overline{AB}+\overline{BC}

eq\overline{AC}$, then point B is not between points A and C" is true. Geometrically, if the sum of the lengths of two line - segments does not equal the length of the third line - segment formed by the same three collinear points, then the middle - point condition is not met.

Answer:

C. If $\overline{AB}+\overline{BC}
eq\overline{AC}$, then point B is not between points A and C is the contra - positive of statement A.