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GEO.25 Use technology to explore conjectures and support proofs Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Geometry Math worksheet provides structured practice on GEO.25 Use technology to explore conjectures and support proofs. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

GEO.25 Use technology to explore conjectures and support proofs - High school geometry skill focused on using technology to explore conjectures and support proofs.

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Grade

Geometry

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related GEO.25 Use technology to explore conjectures and support proofs skill assessments:

Problem 1Difficulty: Mastery

Technology-based exploration: A student explores a 'proof by counterexample' idea using technology. They suspect the claim: 'If two angles in a quadrilateral are congruent, then the quadrilateral is an isosceles trapezoid.' They use a quadrilateral tool to generate many random quadrilaterals with one pair of congruent angles and quickly find an example that is not a trapezoid at all. Which option correctly states (1) the conjecture supported by the exploration and (2) the conclusion that still needs a formal proof?

AConjecture: Finding one valid counterexample is enough to disprove a universal conjecture; technology can help search for counterexamples. — Needs formal proof: The app found one quadrilateral with two equal angles that was not a trapezoid.

BConjecture: If you cannot find a counterexample after many tries, the conjecture is proven. — Needs formal proof: The conclusion 'the conjecture is false for all cases' follows from a single counterexample, but any claim that a conjecture is always true would still require a formal proof; technology alone cannot replace it.

CConjecture: A conjecture is disproved only if you find three counterexamples. — Needs formal proof: The student tested 50 cases, so the result is certain.

DConjecture: Finding one valid counterexample is enough to disprove a universal conjecture; technology can help search for counterexamples. — Needs formal proof: The conclusion 'the conjecture is false for all cases' follows from a single counterexample, but any claim that a conjecture is always true would still require a formal proof; technology alone cannot replace it.

Solution & Explanation:A universal statement ('for all quadrilaterals...') is disproven by one counterexample, and technology is useful for discovering such counterexamples. On the other hand, confirming a universal statement always requires a formal proof, since no finite set of tests can cover all cases.

Problem 2Difficulty: Mastery

Technology-based exploration: A student uses a CAS and graphing tool to explore a coordinate proof idea for a triangle. They place A(0,0), B(1,0), C(u,v) with v > 0. The tool computes the equation of the perpendicular bisector of AB as x = 0.5. It also computes that the circumcenter has x-coordinate 0.5 for many randomly chosen (u,v). Which option correctly states (1) the conjecture supported by the exploration and (2) the conclusion that still needs a formal proof?

AConjecture: The circumcenter always lies on the median from C to AB. — Needs formal proof: To conclude the circumcenter has x = 0.5 for all choices of (u,v), you need a formal proof from the definition of circumcenter as intersection of perpendicular bisectors, not random testing.

BConjecture: In this coordinate setup, the circumcenter always lies on the perpendicular bisector of AB, so its x-coordinate is 0.5. — Needs formal proof: Since AB is symmetric around x = 0.5, the center should be there.

CConjecture: In this coordinate setup, the circumcenter always lies on the perpendicular bisector of AB, so its x-coordinate is 0.5. — Needs formal proof: To conclude the circumcenter has x = 0.5 for all choices of (u,v), you need a formal proof from the definition of circumcenter as intersection of perpendicular bisectors, not random testing.

DConjecture: The circumcenter always has x-coordinate 0.5 for any triangle in the plane. — Needs formal proof: The CAS output repeated x = 0.5 many times, so it must always be true.

Solution & Explanation:The definition of circumcenter guarantees it lies on the perpendicular bisector of AB, which is x = 0.5 in this coordinate placement, so the conjecture is correct. A proof should explicitly use that definition and show it holds for every (u,v) with v > 0, rather than relying on random samples.

Problem 3Difficulty: Mastery

Technology-based exploration: A student uses a dynamic geometry tool to explore whether a conjecture is always true. They construct triangle ABC, then pick a point P on side BC and draw lines through P parallel to AB and AC, forming a small parallelogram-like shape near P. The app shows one pair of opposite sides in the small quadrilateral is always congruent, but the other pair is not unless P is the midpoint of BC. Which option correctly states (1) the conjecture supported by the exploration and (2) the conclusion that still needs a formal proof?

AConjecture: The construction always makes a parallelogram for any point P on BC. — Needs formal proof: In the tested examples, the shape looked most symmetric when P was near the midpoint.

BConjecture: In this construction, a stronger symmetry (like getting a true parallelogram) occurs only at special placements such as P being the midpoint; the general case does not force a parallelogram. — Needs formal proof: To claim 'parallelogram happens only when P is the midpoint' (or any similar necessary-and-sufficient condition), you must provide a formal proof or a counterexample argument, not rely on a few drags.

CConjecture: In this construction, a stronger symmetry (like getting a true parallelogram) occurs only at special placements such as P being the midpoint; the general case does not force a parallelogram. — Needs formal proof: The app showed congruence only for one pair of sides in most drags.

DConjecture: Any quadrilateral with one pair of congruent sides is a parallelogram. — Needs formal proof: To claim 'parallelogram happens only when P is the midpoint' (or any similar necessary-and-sufficient condition), you must provide a formal proof or a counterexample argument, not rely on a few drags.

Solution & Explanation:The exploration indicates that some properties are automatic from the parallel-line construction, while others require special conditions like midpoint placement. Any claim about an exact condition (especially an 'only if') requires formal justification, because dragging cannot check all positions or rule out exceptions.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Geometry Math students. It aligns with educational standard GEO.25 (High school geometry skill focused on using technology to explore conjectures and support proofs.).