Home › Certifications › High School Geometry
150 multiple-choice questions and 165 flashcards, organised into 13 chapters, written to the Common Core HS Geometry standards blueprint. Every question carries a full rationale.
Study High School Geometry in CoStudy →
Common Core HS Geometry standards (HSG-CO, HSG-SRT, HSG-C, HSG-GPE, HSG-GMD, HSG-MG) — covers: congruence, similarity, right triangles and trigonometry, circles, geometric properties with equations, geometric measurement and dimension, modeling with geometry. Includes Euclidean proofs, transformations, coordinate geometry. MCQs reference public Common Core standards.
CoStudy's High School Geometry bank holds 315 items organised into 13 chapters that follow the published blueprint. Every multiple-choice question carries a written rationale explaining why the correct answer is correct and why each distractor is tempting but wrong.
Each chapter follows a domain of the published exam outline. Practise one on its own:
A sample of 12 multiple-choice questions from the bank, with the full rationale shown.
Two triangles are similar by AA (angle-angle). What can we conclude about corresponding sides?
Answer: A — Proportional (ratio of corresponding sides is constant — the scale factor)
Similar triangles: corresponding angles congruent, corresponding sides proportional. Scale factor = ratio of any pair of corresponding sides. AA, SAS, SSS are similarity criteria. Distinct from congruent (same size).
In a 45-45-90 triangle, if the legs equal 1, the hypotenuse equals:
Answer: A — √2
A) Standard ratio 1:1:√2. B/C/D) Each is incorrect.
A rigid motion (translation, rotation, or reflection):
Answer: A — Preserves distance and angle measure
A) Rigid motions (isometries) preserve BOTH distance and angle measure — that's the definition. B) False — angles are preserved. C) False — area is preserved precisely because distances are. D) Describes a dilation (similarity), not a rigid motion.
If two triangles are similar with scale factor k, the ratio of their areas is:
Answer: B — k²
B) Areas scale by k². A) Lengths. C) Volumes. D) Wrong.
A tangent line touches a circle at point P. The angle between the tangent and the radius to P is:
Answer: D — 90°
A) Treats them as the same line — false. B/C) Arbitrary guesses. D) Tangent-Radius Theorem: a tangent line is PERPENDICULAR to the radius drawn to the point of tangency.
Two similar solids have surface areas in ratio 9 : 25. The ratio of their volumes is:
Answer: C — 27 : 125
A) That's the surface-area ratio, not volume. B) Linear scale factor = √(9/25) = 3/5. C) Volumes scale by the CUBE of the linear factor: (3/5)³ = 27/125. D) Squared the surface-area ratio.
Surface area of a cube with edge s is:
Answer: C — 6s² — six faces of area s² each
C) Standard formula. A/B/D) Each is incorrect.
Two triangles have side ratios 2:5. If smaller has area 12, larger has area:
Answer: B — 75 (area scales as square of linear scale; (5/2)² = 25/4; 12 × 25/4 = 75)
Similar figures: area ratio = (linear ratio)². If linear ratio 2:5, area ratio 4:25. Larger area = 12 · (25/4) = 75. Volume scales as cube of linear ratio.
Two triangles are SIMILAR if:
Answer: D — Same shape, proportional sides (corresponding angles equal, corresponding sides in same ratio)
Similar: same shape, sides proportional. Proof: AA (two angles equal), SAS similarity (one angle equal + sides around proportional), SSS similarity (all sides proportional). Scale factor preserves ratios.
The midpoint of (2, 4) and (8, 10) is:
Answer: B — (5, 7) — midpoint = ((2+8)/2, (4+10)/2) = (5, 7)
Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2). Average of x's, average of y's. (5, 7). Used for bisecting segments. Coordinate geometry basics.
Tangent to a circle is _____ to the radius at the point of tangency.
Answer: C — Perpendicular (tangent ⊥ radius at point of tangency)
Key circle theorem: tangent meets radius at 90°. Used in many proofs and right-triangle problems involving circles. G-C standards.
A cone has radius 3 and height 4. Its volume is:
Answer: A — 12π
A) V = (1/3)πr²h = (1/3)π(9)(4) = 12π. B) Forgot the 1/3 — that's the cylinder volume. C) Multiplied wrong factor. D) Used just r² without h.
6 sample cards from the 165 in the bank.
In a right triangle with hypotenuse 13 and leg 5, find the other leg.
12. (169 − 25 = 144; √144 = 12.)
What is sin(45°)?
√2/2 ≈ 0.707.
Sum of exterior angles of any polygon (one at each vertex)?
Always 360°.
Reflect (4, 7) across the x-axis. Result?
(4, −7).
What is the 'inverse sine' (sin⁻¹) used for?
Finding the angle when you know the ratio. Like sin⁻¹(0.5) = 30°.
What's the relationship between a tangent and the radius at the point of tangency?
Perpendicular (90°).
These samples are a small slice. The full bank runs flashcards, multiple choice and timed mock exams with per-chapter progress tracking, on the web and in the iOS app.
The High School Geometry bank holds 315 items: 150 multiple-choice questions, 165 flashcards. 18 of them are on this page to read free, with no signup.
Yes. Every multiple-choice item carries a written rationale that states the controlling principle behind the correct answer and then addresses each wrong option in turn — why it tempts and precisely where it fails. Knowing why the plausible answer was wrong is worth more than knowing which letter was right.
It is organised into 13 chapters that follow the published exam blueprint: Tools of Geometry: Points, Lines, Planes, Angles; Reasoning and Proof; Parallel and Perpendicular Lines; Congruent Triangles; Relationships Within Triangles; Quadrilaterals and Polygons; Similarity; Right Triangles and Trigonometry; Circles; Area and Perimeter; Surface Area and Volume; Transformations and Symmetry; Probability and Geometry. The number of questions in each chapter is proportional to that domain's published weight, so working through the bank exposes you to roughly the mix the real exam uses.
Common Core HS Geometry standards (HSG-CO, HSG-SRT, HSG-C, HSG-GPE, HSG-GMD, HSG-MG) — covers: congruence, similarity, right triangles and trigonometry, circles, geometric properties with equations, geometric measurement and dimension, modeling with geometry. Includes Euclidean proofs, transformations, coordinate geometry. MCQs reference public Common Core standards.
The samples on this page are free to read in full, rationales included, with no account. The complete 315-item bank, the timed mock exams and per-chapter progress tracking are part of CoStudy on the web and in the iOS app.
Last reviewed 2026-08-22. Banks are written against the certifying body's published exam outline and re-checked when that outline changes — exams get renumbered, retired and reweighted, and a bank written to a superseded outline teaches the wrong proportions. Figures that are re-indexed annually are deliberately not asserted as rules; the questions test the governing principle instead.