Undefined Terms & Postulates
Name two undefined terms in geometry
Example answers: point, line, plane
Which of these is an example of a defined term: midpoint, angle, or perpendicular?
100 — Example: midpoint is a defined term (midpoint defined using segment and equality of segments).
Are the sides of a spherical triangle straight lines or arcs?
100 — H. The sides are arcs of great circles
100 — Give a quick counterexample to the statement: "If a number is a perfect square, then the number is even."
Example: 9 is a perfect square (3^2) but 9 is odd.
100 — In the provided sphere picture, identify whether curve q is a great circle arc (spherical line), a small circle, or neither.
100 — q = spherical line (great circle arc).
Which statement describes: "If point B is on AC and between A and C, then AB + BC = AC"? (
A. Segment addition postulate.
Name the property or theorem that justifies congruent segments can be added to congruent segments to produce congruent sums
200 — Addition property of equality or segment addition (teacher may accept segment addition postulate for segments).
True or False: Spherical triangles can have angle measures summing to exactly 180°
200 — False. Spherical triangle angle sums are greater than 180° (and less than 540°)
200 — Provide a counterexample to: "For every integer n, n^3 is positive."
200 — Example: n = -1 gives n^3 = -1, which is not positive.
200 — Identify curve p (from the same picture)
200 — p = not a great circle, it is a small circle
Give the definition of a postulate and give one example used in Euclidean geometry about points
Postulate: an accepted statement assumed true without proof. Example: Through any two distinct points there is exactly one line
True or False: In Euclidean geometry, parallel lines never meet. Explain with one sentence
300 — True. In Euclidean geometry parallel lines are defined as coplanar lines that do not intersect.
Which statements are true about spherical triangles? (Select TWO)
A. sum is greater than 180° and less than 540°
B. cannot have angles >90°
C. sides are arcs of great circles are arcs;
D. sum always 180°
E. must have at least one right angle.
A. sum is greater than 180° and less than 540°
C. sides are arcs of great circles are arcs;
300 — Explain why a single counterexample is enough to disprove a universal mathematical statement
300 — Because universal statements claim something for all elements; one counterexample shows the claim is false
300 — On a labeled figure, name r.
Great Circle R
Explain why undefined terms are important when defining other geometric terms. (1 sentence)
Undefined terms form the basic vocabulary so definitions aren't circular; they let postulates and theorems be built consistently
Provide a counterexample to disprove: "If a four-sided shape has two sides of equal length, then it must be a rectangle."
400 — A kite or isosceles trapezoid can show two sides equal but not a rectangle; answer:
Explain in 1–2 sentences how parallel lines differ on a sphere versus in Euclidean plane geometry
400 — On a sphere, "parallel lines" (distinct great circles) always intersect; there are no Euclidean parallels — great circles intersect in two antipodal points.
400 — Construct a short proof or counterexample to show whether "All rectangles are squares" is true or false
400 — False. Counterexample: 2 by 4 rectangle is not a square because adjacent sides differ.
Squares are also quadrilaterals with FOUR equal sides, and rectangles only have 2
400 — Can spherical lines be parallel? Choose: Yes / No. Support your answer with one-sentence
400 — No. Spherical lines (great circles) always intersect