Axioms, segments and theorems
See that one line passes through two points and that AB = AC + CB, find the midpoint of a segment and tell a theorem from its converse.
- Ukraine: grade 7
Guide
Ukrainian Geometry 7 (§§ 1, 2, 6, 12) teaches these: points and lines, the length of a segment, axioms and theorems.
Basic facts about points and lines: through any two points there passes a line, and only one; two different lines have at most one common point. Statements accepted without proof are called axioms.
If C lies between A and B on the segment AB, then AB = AC + CB. The midpoint is the point M with AM = MB, so AM = AB : 2.
A theorem is a statement that is proved. It has a condition (what is given) and a conclusion (what is to be proved). Swap them and you get the converse theorem. A converse is not always true.
If a theorem and its converse are both true, the condition is necessary and sufficient for the conclusion. Drag the points to see what always stays true.