Use incidence, betweenness and definitions to justify geometric claims.
Loading your saved responses…
On a narrow screen, scroll diagrams horizontally to read every label. Keyboard users can focus a diagram and use the arrow keys.
Try
Draw three points A, B and C on one line with B between the others. If AB=3 cm and BC=5 cm, the total AC is 8 cm. Now move B off the line while keeping AB and BC fixed. The sum of the two short lengths is still 8, but it no longer necessarily equals AC.
Predict
Predict which part of the first description made addition legitimate. Geometry starts from agreed basic assumptions, called axioms or postulates; definitions specify what words mean; theorems are deductions from those assumptions. In Euclidean geometry, two distinct points determine one line. A segment is the part between its endpoints. “Between” includes collinearity and order.
Experiment
Inspect the diagram, then redraw the three points in a different orientation. Mark only information explicitly given: equality ticks for known equal segments, a square for a known right angle, and arrows for given parallel lines. A mark records an assumption or established result, not a feature you guessed from appearance.
Measured line segment: AB=3 units, BC=5 units and AC=8 units. B lies between A and C; the drawing uses one common length scale.
Understand
Worked argument: let M be the midpoint of AB, where AB=10. By definition M lies on AB and AM=MB. Segment addition gives AM+MB=10, so 2AM=10 and AM=5. The two premises—membership and equality—play different roles. A point away from AB can be equidistant from A and B without being its midpoint.
Invalid proof: “The two pieces look equal, so M is the midpoint.” A drawing alone establishes neither exact equality nor membership. Repair it by naming the given midpoint or proving both defining conditions. In written arguments, state the source of each fact: given, definition, axiom, or earlier theorem.
Apply
Use the practice to turn a visual claim into a short chain of justified statements. A counterexample must meet every stated hypothesis. A bent path is a valid objection to unrestricted segment addition, but not to the theorem that already assumes betweenness.
Continue
Depth: axiomatic systems can use different starting assumptions while describing the same Euclidean geometry. This unit uses familiar incidence, order, distance and angle facts rather than developing a complete formal axiom system. Continue to angle arguments, where subtracting the same adjacent angle will prove a visual symmetry.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
B lies between A and C on a line. AB=7/2 cm and BC=9/2 cm. Find AC in cm.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
M is the midpoint of PQ and PQ=15 cm. Find PM in cm.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which is a definition rather than a conclusion inferred from appearance?
Which assumption is needed to infer AC=AB+BC?
On segment AB of positive length, suppose M and N are both midpoints. Prove M=N using distance from A along the segment.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A learner says AB=3 and BC=4 imply AC=7 for any three points. Give a geometric counterexample and state a repaired theorem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
