The Life Starter
Sign up
Grade 10 · unit review

Review: Geometry: Claims, Converses, and Proof

Geometry arguments connect definitions, givens and established properties. A conclusion needs a chain of reasons; an attractive diagram may suggest a relation but does not establish every length, angle or universal claim.

Download reviews and answer keys

Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. A fictional geometry rule says: If a quadrilateral is a square, it is a rectangle. Figure 7 is a square. What follows from this rule?
Need a hint?
  • An implication allows its conclusion when its hypothesis holds.
  • Track the exact direction of the rule.
2. Write the converse: If a point is on a perpendicular bisector, then it is equidistant from the segment endpoints.
Need a hint?
  • A converse swaps an implication’s hypothesis and conclusion.
  • Reversing a statement differs from negating its parts.
3. Among candidate rectangles with listed side lengths including 7 and 8, which refutes “Every rectangle has four equal side lengths”?
Need a hint?
  • A universal claim fails if one permitted example satisfies its hypothesis but violates its conclusion.
  • A counterexample must belong to the group named in the claim.
4. Given AB=BC, BC=CD, and AB=10 cm, find CD.
Need a hint?
  • If a first segment equals a second and that second equals a third, transitivity establishes equality of the first and third.
  • Only supplied or proved equalities can enter the chain.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. A fictional geometry rule says: If a quadrilateral is a square, it is a rectangle. Figure 9 is a square. What follows from this rule? Candidate answers: (A) Every quadrilateral is a square (B) Every rectangle is a square (C) Figure cannot have equal sides (D) Figure is a rectangle. 2. Write the converse: If an angle is a right angle, then its measure is ninety degrees. Candidate answers: (A) If an angle is a right angle, then not (its measure is ninety degrees). (B) If its measure is ninety degrees, then an angle is a right angle. (C) If not (an angle is a right angle), then not (its measure is ninety degrees). (D) If not (its measure is ninety degrees), then an angle is a right angle.. 3. Among candidate rectangles with listed side lengths including 9 and 10, which refutes “Every rectangle has four equal side lengths”? Candidate answers: (A) 10 by 10 (B) 11 by 11 (C) 9 by 10 (D) 9 by 9. 4. Given AB=BC, BC=CD, and AB=12 cm, find CD. Candidate answers: (A) 11 cm (B) 12 cm (C) 13 cm (D) 24 cm.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. Figure is a rectangle. The premise matches the stated sufficient condition, so the conclusion follows. The statement does not establish the reverse implication or make all quadrilaterals rectangles. 2. If its measure is ninety degrees, then an angle is a right angle.. A converse swaps hypothesis and conclusion. It creates a new statement whose truth needs its own justification; writing the reverse does not by itself prove it. 3. 9 by 10. A 9-by-10 rectangle has four right angles and therefore satisfies the hypothesis, but unequal adjacent sides because 9≠10. One counterexample refutes a universal claim. 4. 12 cm. Transitivity gives AB=CD through their shared equality to BC. Since AB=12 cm, CD=12 cm. The givens establish this value; no measurement from a sketch is needed.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.