Write the converse of: If an angle is right, then it measures 90°.
Answer: If an angle measures 90°, then it is right.
Reasoning: Interchange hypothesis and conclusion.
Reasoning: Interchange hypothesis and conclusion.
Unit 01 · Lesson 05 · Foundations, Constructions & Geometric Reasoning
Conditional Statements, Converses & Counterexamples · Teacher Key
Write conditional, converse, inverse, and contrapositive statements; evaluate truth and produce counterexamples.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to write conditional, converse, inverse, and contrapositive statements; evaluate truth and produce counterexamples.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Use definitions, measurements, constructions, and logical statements to establish a precise geometric argument.
AB=|b-a|M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)m\angle A+m\angle B=180^\circ
Worked reasoning
Process: State what is given, translate each fact into a definition or theorem, and justify every conclusion in order.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: A diagram suggests relationships, but only stated markings, definitions, and proven results may be used as evidence.
Khaemenes Academy · Open-Age Geometry
Conditional Statements, Converses & Counterexamples
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Give a counterexample to: All quadrilaterals with four equal sides are squares.
Answer: A non-square rhombus
Reasoning: A rhombus may have non-right angles.
Reasoning: A rhombus may have non-right angles.
State the contrapositive of: If two lines are parallel, then they do not intersect.
Answer: If two lines intersect, then they are not parallel.
Reasoning: Negate and reverse.
Reasoning: Negate and reverse.
Is the inverse of a true conditional always true?
Answer: No
Reasoning: A conditional and its inverse are not logically equivalent.
Reasoning: A conditional and its inverse are not logically equivalent.
Translate “p only if q” into implication notation.
Answer: p → q
Reasoning: The phrase 'only if' introduces the necessary condition.
Reasoning: The phrase 'only if' introduces the necessary condition.
What single counterexample proves about a universal statement?
Answer: It is false
Reasoning: One valid exception disproves a claim about all cases.
Reasoning: One valid exception disproves a claim about all cases.