What are common Pythagorean triples?
Answer: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples.
Explanation
Save time on tests.
Explore · Flashcard deck
Geometry: Proofs, Triangles, and Circles (High School)
Right trianglesTransformationsCirclesPolygonsQuadrilateralsLogicTrianglesAngles
Shared from Quizcard. Flip through the cards below, or copy the deck to edit and study it in your own account — free.
Space flips · arrows move · Esc exits
Right triangles
In a right triangle, what does the altitude to the hypotenuse create?
Answer: Three similar triangles; the altitude is the geometric mean of the two hypotenuse segments.
Explanation
Each leg is the geometric mean of the hypotenuse and the adjacent segment.
State the Pythagorean theorem and its converse.
Answer: a² + b² = c² in a right triangle; if a² + b² = c² for a triangle's sides, it is a right triangle.
Explanation
If a² + b² < c², the triangle is obtuse; > means acute.
Transformations
How does a 90° counterclockwise rotation about the origin change coordinates?
Answer: (x, y) → (−y, x)
Explanation
180°: (−x, −y). 90° clockwise: (y, −x).
How does a reflection over the x-axis change coordinates?
Answer: (x, y) → (x, −y)
Explanation
Over the y-axis: (−x, y). Over y = x: (y, x).
Which transformations are rigid (preserve size and shape)?
Answer: Translations, rotations, and reflections.
Explanation
Dilations change size but preserve shape (similarity).
Circles
What does the chord-chord power theorem say?
Answer: If two chords intersect inside a circle, the products of their segments are equal: a·b = c·d.
Explanation
Secant-secant: (whole)(outside) = (whole)(outside).
Arc length and sector area formulas
Answer: Arc length = (θ/360°) · 2πr; sector area = (θ/360°) · πr²
Explanation
θ is the central angle in degrees.
What is the relationship between a tangent and the radius at the point of tangency?
Answer: They are perpendicular.
Explanation
Two tangent segments from the same external point are congruent.
What is true of an angle inscribed in a semicircle?
Answer: It is a right angle (Thales' theorem).
Explanation
Opposite angles of an inscribed quadrilateral are supplementary.
What is the relationship between a central angle and its arc?
Answer: The central angle equals the measure of its intercepted arc.
Explanation
An inscribed angle is half the intercepted arc.
Polygons
How many diagonals does an n-sided polygon have?
Answer: n(n − 3)/2
Explanation
A hexagon has 9 diagonals.
What is the sum of exterior angles of any convex polygon?
Answer: 360°
Explanation
Each exterior angle of a regular n-gon is 360°/n.
Quadrilaterals
What is the area of a rhombus or kite using diagonals?
Answer: A = ½ · d1 · d2
Explanation
Works because the diagonals are perpendicular.
How do rectangles, rhombi, and squares differ?
Answer: Rectangle: four right angles, congruent diagonals. Rhombus: four congruent sides, perpendicular diagonals that bisect angles. Square: both.
Explanation
A square is both a rectangle and a rhombus.
What are the properties of a parallelogram?
Answer: Opposite sides parallel and congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other.
Explanation
Rectangles, rhombi, and squares are special parallelograms.
Logic
What is a conditional statement and its converse?
Answer: If p then q; the converse swaps them: if q then p.
Explanation
The converse is not always true.
What is a biconditional?
Answer: A statement true in both directions: p if and only if q.
Explanation
Definitions are biconditionals.
What is the contrapositive and why does it matter?
Answer: If not q then not p; it is logically equivalent to the original conditional.
Explanation
Proving the contrapositive proves the statement.
Triangles
What is the hinge theorem?
Answer: If two triangles have two congruent sides, the one with the larger included angle has the longer third side.
Explanation
Also called SAS inequality.
What are the centroid, circumcenter, incenter, and orthocenter?
Answer: Centroid: medians meet (balance point, 2:1 ratio). Circumcenter: perpendicular bisectors meet (equidistant from vertices). Incenter: angle bisectors meet (equidistant from sides). Orthocenter: altitudes meet.
Explanation
The circumcenter is the center of the circumscribed circle.
What is the triangle midsegment theorem?
Answer: The segment joining the midpoints of two sides is parallel to the third side and half its length.
Explanation
Useful in coordinate proofs.
What does the exterior angle theorem say?
Answer: An exterior angle of a triangle equals the sum of the two remote interior angles.
Explanation
Follows from the 180° angle sum.
What is the isosceles triangle theorem?
Answer: If two sides of a triangle are congruent, the angles opposite them are congruent (and conversely).
Explanation
Base angles are equal.
What are the triangle similarity criteria?
Answer: AA, SAS (proportional sides with included angle), SSS (all sides proportional).
Explanation
Similar triangles have proportional sides and equal angles.
What is CPCTC?
Answer: Corresponding Parts of Congruent Triangles are Congruent, used after proving triangles congruent.
Explanation
A common final step in proofs.
What are the triangle congruence postulates?
Answer: SSS, SAS, ASA, AAS, and HL (right triangles).
Explanation
SSA and AAA do not prove congruence.
Angles
What is the angle bisector theorem?
Answer: A point on an angle's bisector is equidistant from the sides of the angle.
Explanation
Perpendicular bisector: equidistant from the endpoints of a segment.
When a transversal crosses parallel lines, which angle pairs are congruent?
Answer: Corresponding angles, alternate interior angles, and alternate exterior angles.
Explanation
Same-side (consecutive) interior angles are supplementary.
What are vertical angles?
Answer: Opposite angles formed by two intersecting lines; they are congruent.
Explanation
Adjacent angles on a line are supplementary (linear pair).
Make your own deck like this
Upload a PDF, Word doc, or slides and Quizcard writes the cards for you, or build them by hand. Free to start, no credit card required.