Geometry foundations: calculation and proof assessment
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 90 minutes, as an unvalidated planning estimate. This collection contains 8 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the set before revealing criteria or worked solutions. Choose your own methods and show the intermediate reasoning on paper or in the written responses.
Check domains, hypotheses and units. An exact answer match only checks the requested value; written work needs review.
Suggested time is guidance, not an access limit. First responses and known answer guidance remain in saved history. These public tasks do not certify independent mastery or readiness for a whole course examination.
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0 of 8 tasks have recorded responses. No overall mastery score is assigned.
A transversal meets two distinct coplanar lines. A pair of alternate interior angles measure (3x+7)° and (5x−21)°. If the lines are parallel, determine x and those angles. Conversely, if the two angles are known equal without assuming parallelism, identify the theorem that proves parallelism. Explain why appearance alone would not suffice.
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.
Triangle ABC has AB=AC. M is the midpoint of BC. Prove triangles ABM and ACM congruent and prove AM perpendicular to BC. Does this alone imply AM is an angle bisector at B? Explain.
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.
Triangle vertices are A=(1,1), B=(4,1), C=(1,5). Reflect across the y-axis, then dilate by factor 2 about the origin. Give the final vertices, compare side lengths and area with the original, and decide whether the final triangle is congruent or merely similar to the original.
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 triangle has side lengths 7, 9 and 12. Compute 12²−(7²+9²). Use the sign on paper to classify the angle opposite the longest side as acute, right or obtuse. Enter the computed difference.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A vertical mast stands on level ground. From a ground-level point 24 m horizontally from its base, tan θ=5/12 for the angle of elevation to its top. Find the height and line-of-sight distance. If the observation point moves to 12 m from the base, find the new tangent ratio and explain why the elevation angle does not simply double.
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 quadrilateral has vertices (0,0), (8,0), (6,4), (0,4) in that order. Prove that it is a trapezoid, find its area by decomposition, and calculate the magnitude sum of its exterior turns when traversed once. Explain why a self-intersecting order would need separate treatment.
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 circle has center O and radius 9 cm. From point P with OP=15 cm a tangent touches at T. Find PT and justify the needed right angle. A separate chord AB subtends a central minor angle AOB of 110°. Find an inscribed angle ACB when C lies on the major arc AB, stating which arc it intercepts.
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.
Points A=(0,0), B=(6,0) and C=(2,4) form a triangle. Describe a compass-and-straightedge construction of their circumcenter using perpendicular bisectors. Find its coordinates and radius, and prove the resulting circle passes through all three points. Explain why no circle can pass through three distinct collinear points.
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.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A transversal meets two distinct coplanar lines. A pair of alternate interior angles measure (3x+7)° and (5x−21)°. If the lines are parallel, determine x and those angles. Conversely, if the two angles are known equal without assuming parallelism, identify the theorem that proves parallelism. Explain why appearance alone would not suffice.
Triangle ABC has AB=AC. M is the midpoint of BC. Prove triangles ABM and ACM congruent and prove AM perpendicular to BC. Does this alone imply AM is an angle bisector at B? Explain.
Triangle vertices are A=(1,1), B=(4,1), C=(1,5). Reflect across the y-axis, then dilate by factor 2 about the origin. Give the final vertices, compare side lengths and area with the original, and decide whether the final triangle is congruent or merely similar to the original.
A triangle has side lengths 7, 9 and 12. Compute 12²−(7²+9²). Use the sign on paper to classify the angle opposite the longest side as acute, right or obtuse. Enter the computed difference.
A vertical mast stands on level ground. From a ground-level point 24 m horizontally from its base, tan θ=5/12 for the angle of elevation to its top. Find the height and line-of-sight distance. If the observation point moves to 12 m from the base, find the new tangent ratio and explain why the elevation angle does not simply double.
A quadrilateral has vertices (0,0), (8,0), (6,4), (0,4) in that order. Prove that it is a trapezoid, find its area by decomposition, and calculate the magnitude sum of its exterior turns when traversed once. Explain why a self-intersecting order would need separate treatment.
A circle has center O and radius 9 cm. From point P with OP=15 cm a tangent touches at T. Find PT and justify the needed right angle. A separate chord AB subtends a central minor angle AOB of 110°. Find an inscribed angle ACB when C lies on the major arc AB, stating which arc it intercepts.
Points A=(0,0), B=(6,0) and C=(2,4) form a triangle. Describe a compass-and-straightedge construction of their circumcenter using perpendicular bisectors. Find its coordinates and radius, and prove the resulting circle passes through all three points. Explain why no circle can pass through three distinct collinear points.
