108 cards

Geometry Flash Cards: Angle Pairs, Theorems and Formulas

Name the term or the formula, turn the card to check, then print a set on one sheet.

Between the lines, opposite sides

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Type the word, or the formula as in pi r^2 or 1/2bh

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How to practise

Next step Name the angle pair, then find x: 8 figures with a key Next: geometry words in a word search

All 108 cards

Angle pairs and lines 12

A line that crosses two others Transversal: it cuts each line at its own point. Across two lines it makes eight angles.
Adjacent angles forming a line Linear pair: two adjacent angles whose outer sides make a straight line. They add to 180°.
Between the lines, opposite sides Alternate interior angles: between the lines, on opposite sides. Equal when lines are parallel.
Crosses a segment's midpoint at 90° Perpendicular bisector: every point on it is the same distance from the segment's two ends.
Lines that cross at 90° Perpendicular lines cross at right angles, so all four angles at the crossing are 90°.
Opposite angles at a crossing Vertical angles face each other across the crossing point, and they are always equal.
Outside the lines, opposite sides Alternate exterior angles: outside the lines, on opposite sides. Equal when lines are parallel.
Ray that cuts an angle in half Angle bisector: a ray from the vertex that splits an angle into two equal angles.
Same corner at both crossings Corresponding angles sit in the same corner at each crossing. Parallel lines make them equal.
Share a vertex and one side Adjacent angles share a vertex and one side, and neither lies inside the other.
Two angles that add to 90° Complementary angles add to 90°, like 30° and 60°. Each is the other's complement.
Two angles that add to 180° Supplementary angles add to 180°, like 110° and 70°. They do not have to touch.

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Triangle facts 12

All 3 sides of different lengths Scalene triangle: three sides of different lengths, so three different angles.
Any 2 sides added beat the third Triangle inequality: any two sides add to more than the third. 2, 3 and 6 make no triangle.
Equals the 2 remote interior angles Exterior angle: it equals the sum of the two interior angles that do not touch it.
Isosceles: the base angles are Equal: the angles opposite the two equal sides of an isosceles triangle always match.
Pythagorean theorem a² + b² = c² in a right triangle with legs a and b and hypotenuse c. Try 3, 4, 5: 9 + 16 = 25.
Right triangle: x, x, x√2 45-45-90: two equal legs, and the hypotenuse is a leg times √2, about 1.414.
Right triangle: x, x√3, 2x 30-60-90: the hypotenuse is twice the short leg, and the long leg is the short leg times √3.
Side opposite the right angle Hypotenuse: always the longest of the three sides, and the c in a² + b² = c².
Sum of a triangle's angles 180°: the three angles of any triangle add to a straight angle.
Triangle: 2 sides the same length Isosceles triangle: two equal sides, and the two angles opposite them are equal too.
Vertex to the opposite midpoint Median: from a vertex to the midpoint of the opposite side. All three meet at one point.
Vertex to the opposite side at 90° Altitude: its length is the height of the triangle. In an obtuse triangle it can fall outside.

Congruence and similarity 12

2 angles and a side not between AAS: two angles and a side not between them match, so the triangles are congruent.
2 angles and the side between ASA: two angles and the side between them match, so the triangles are congruent.
2 sides and the angle between SAS: two sides and the angle between them match, so the triangles are congruent.
3 pairs of equal sides SSS: three sides of one triangle equal three sides of another, so the triangles are congruent.
CPCTC: matching parts are Congruent, known as CPCTC: corresponding parts of congruent triangles are congruent.
Equal angles, sides in proportion Similar figures have the same shape, in any size: one is a scaled copy of the other.
Fails: 2 sides, angle not between SSA proves nothing: two sides and an angle not between them can fit two different triangles.
Fits exactly on top of another Congruent figures have the same shape and size: slides, flips and turns match them up.
Ratio of matching sides Scale factor: the ratio of matching lengths in similar figures. Sides 3 and 6 give 2.
Right triangles: hypotenuse, 1 leg HL: two right triangles with equal hypotenuses and one equal leg are congruent.
Scale factor k: areas multiply by k²: double every length and the area is 4 times as large; triple it and the area is 9 times.
Similar by 2 equal angles AA: two angles of one triangle equal two angles of another, so the triangles are similar.

Quadrilaterals by property 12

2 pairs of equal adjacent sides Kite: two pairs of equal sides next to each other. Its diagonals cross at right angles.
All 4 corners lie on one circle Cyclic quadrilateral: all four vertices on one circle. Its opposite angles add to 180°.
Both a rectangle and a rhombus Square: the one quadrilateral with four equal sides and four equal angles.
Diagonals bisect each other Parallelogram: the diagonals bisect each other, and opposite sides are equal and parallel.
Equal sides, diagonals at 90° Rhombus: four equal sides. Its diagonals cross at right angles and cut each other in half.
Joins 2 corners, not neighbors Diagonal: it joins two vertices of a polygon that do not share a side.
Joins a trapezoid's leg midpoints Midsegment, or median: parallel to the two parallel sides and as long as their average.
Parallelogram with equal diagonals Rectangle: a parallelogram is a rectangle exactly when its two diagonals are the same length.
Parallelogram: adjacent angles sum 180°: next-door angles of a parallelogram are supplementary, like 70° and 110°.
Parallelogram: opposite angles are Equal: in a parallelogram each angle matches the angle across from it.
Sum of a quadrilateral's angles 360°: one diagonal splits a quadrilateral into two triangles, and 2 × 180° = 360°.
Trapezoid with equal legs Isosceles trapezoid: equal legs, equal base angles and equal diagonals.

Polygons 12

All sides equal, all angles equal Regular polygon: every side the same length and every angle the same size.
Diagonals in a polygon, n sides n(n − 3)/2: each of the n corners joins n − 3 others, and that counts every diagonal twice.
Each angle of a regular hexagon 120°: the six angles add to 4 × 180° = 720°, and 720° ÷ 6 = 120°.
Each angle of a regular octagon 135°: the eight angles add to 6 × 180° = 1080°, and 1080° ÷ 8 = 135°.
Each angle of a regular pentagon 108°: the five angles add to 3 × 180° = 540°, and 540° ÷ 5 = 108°.
Has an interior angle over 180° Concave polygon: at least one interior angle is more than 180°, so a corner points inward.
Polygon with 7 sides Heptagon: 7 sides and 7 angles. Hepta is Greek for seven.
Polygon with 9 sides Nonagon, also called an enneagon. Each angle of a regular nonagon is 140°.
Polygon with 10 sides Decagon: its angles add to 1440°, so each angle of a regular decagon is 144°.
Polygon with 12 sides Dodecagon: 12 sides and 12 angles. Each angle of a regular dodecagon is 150°.
Sum of a polygon's exterior angles 360°: one exterior angle at each corner of a convex polygon. Together they make a full turn.
Sum of interior angles, n sides (n − 2) × 180°: n sides cut into n − 2 triangles. A hexagon has 4 × 180° = 720°.

Circle parts and rules 12

An inscribed angle is ___ its arc Half: an inscribed angle is half the central angle on the same arc. An 80° arc gives 40°.
Angle between tangent and radius 90°: a tangent is perpendicular to the radius drawn to the point where it touches.
Angle inscribed in a semicircle 90°: an inscribed angle whose sides pass through the two ends of a diameter is a right angle.
Angle with its vertex at the center Central angle: vertex at the center, sides along radii. Twice any inscribed angle on its arc.
Angle with its vertex on the circle Inscribed angle: its vertex is on the circle and its sides are chords.
Chord through the center Diameter: a chord through the center, twice as long as the radius.
Circumference ÷ diameter π: the same ratio for every circle, about 3.14159.
Line cutting a circle at 2 points Secant: a line that crosses a circle at two points. The part inside the circle is a chord.
Line touching a circle at 1 point Tangent: it meets the circle once, at the point of tangency, and never goes inside.
Part of a circle between 2 points Arc: a connected piece of a circle. Two points on a circle cut it into two arcs.
Segment with both ends on a circle Chord: a segment with both ends on the circle. The longest chord is a diameter.
Slice between 2 radii and an arc Sector: shaped like a slice of pie, with the center of the circle at its tip.

Area and circle formulas 12

Arc length, angle θ° (θ/360)2πr: the fraction θ/360 of the circumference. A 90° arc is a quarter of 2πr.
Area of a circle πr²: π times the radius squared. A circle of radius 1 has area π, about 3.14.
Area of a parallelogram bh: base times height. The height is the distance between the base and the opposite side.
Area of a rectangle lw: length times width. A 3 by 4 rectangle covers 12 unit squares.
Area of a regular polygon ½ap: half the apothem a (center to the middle of a side) times the perimeter p.
Area of a rhombus or a kite ½d₁d₂: half the product of the two diagonals, which cross at right angles.
Area of a sector, angle θ° (θ/360)πr²: the fraction θ/360 of the whole circle's area. A 90° sector is a quarter of πr².
Area of a trapezoid ½(b₁ + b₂)h: the average of the two bases times the height between them.
Area of a triangle ½bh: half of base times height, where the height meets the base at a right angle.
Area of an equilateral triangle (√3/4)s², where s is the side. A side of 2 gives √3, about 1.73.
Circumference of a circle 2πr, which is also πd: the distance round a circle of radius r and diameter d.
Triangle area: sides a, b, angle C ½ab sin C: half the product of two sides and the sine of the angle between them.

Volume and surface area 12

Surface area of a cone πr² + πrl: the circle base plus the curved side, where l is the slant height.
Surface area of a cube, edge s 6s²: six square faces, each of area s². An edge of 2 gives 24.
Surface area of a cylinder 2πr² + 2πrh: two circle ends plus the side, which unrolls into a rectangle 2πr by h.
Surface area of a sphere 4πr²: four times the area of a circle with the same radius.
Surface area, rectangular prism 2(lw + lh + wh): three pairs of matching faces. A 2 by 3 by 4 box has 52 square units.
Volume of a cone ⅓πr²h: one third of the cylinder with the same base and height.
Volume of a cube, edge s s³: edge times edge times edge. An edge of 2 gives 8.
Volume of a cylinder πr²h: the area of the circle base, πr², times the height h.
Volume of a pyramid ⅓Bh: one third of base area times height, a third of the prism with that base and height.
Volume of a rectangular prism lwh: length times width times height. A 2 by 3 by 4 box holds 24 unit cubes.
Volume of a sphere (4/3)πr³: four thirds of π times the radius cubed. Radius 1 gives about 4.19.
Volume of any prism Bh: the area B of the base times the height h. A cylinder follows the same rule.

Transformations 12

Enlarge or shrink from a center Dilation: distances from the center are multiplied by the scale factor. Angles do not change.
Flip a figure over a line Reflection: each point crosses the mirror line to the same distance on the other side.
Moves that keep every length Rigid motions: translations, reflections and rotations. They keep lengths and angles.
Reflect (x, y) over the line y = x (y, x): the two coordinates trade places. The point (3, 4) goes to (4, 3).
Reflect (x, y) over the x-axis (x, −y): x stays and y changes sign. The point (3, 4) goes to (3, −4).
Reflect (x, y) over the y-axis (−x, y): y stays and x changes sign. The point (3, 4) goes to (−3, 4).
Rotate (x, y) 90° counterclockwise (−y, x): a quarter turn to the left about the origin. The point (3, 4) goes to (−4, 3).
Rotate (x, y) 180° (−x, −y): a half turn about the origin changes both signs. (3, 4) goes to (−3, −4).
Slide every point the same way Translation: every point moves the same distance in the same direction. Nothing turns or flips.
Translate (x, y) right a and up b (x + a, y + b): add a to every x and b to every y. Right 2, up 5 takes (3, 4) to (5, 9).
Turn a figure about a point Rotation: every point turns through the same angle about one fixed center.
What a dilation does not change Angles: a dilation changes lengths by the scale factor and leaves every angle as it was.

How this list was compiled: Own work: 108 cards written for this site. The choice of cards, the wording and the worked examples are ours. Each word, rule and formula was read on 2026-10-01 in two references. One is Wikipedia (https://en.wikipedia.org/wiki/ plus the title): Angle, Transversal (geometry), Bisection, Perpendicular, Triangle, Exterior angle theorem, Pythagorean theorem, Hypotenuse, Special right triangle, Triangle inequality, Median (geometry), Altitude (triangle), Isosceles triangle, Congruence (geometry), Similarity (geometry), Quadrilateral, Rhombus, Rectangle, Parallelogram, Kite (geometry), Trapezoid, Isosceles trapezoid, Cyclic quadrilateral, Internal and external angles, Diagonal, Concave polygon, Regular polygon, Heptagon, Nonagon, Decagon, Dodecagon, Circle, Inscribed angle, Thales's theorem, Tangent lines to circles, Central angle, Circular sector, Area, Equilateral triangle, Volume, Surface area, Cube, Cylinder, Cone, Pyramid (geometry), Sphere, Translation (geometry), Reflection (mathematics), Rotation (mathematics), Homothety and Rigid transformation. The other is the American Heritage Dictionary of the English Language (https://www.ahdictionary.com/word/search.html?q= plus the word) for the words it defines, such as complementary angles, vertical angle, transversal, hypotenuse, scalene, median, rhombus, kite, apothem and sector; Wiktionary (https://en.wiktionary.org/wiki/ plus the word) for linear pair, corresponding angle, CPCTC, cyclic quadrilateral, isosceles trapezoid and concave; Wolfram MathWorld for the inscribed angle and the perpendicular bisector (https://mathworld.wolfram.com/InscribedAngle.html , https://mathworld.wolfram.com/PerpendicularBisector.html ); and the Common Core State Standards for Mathematics (https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf ) for the theorems they name: the angle sum and exterior angle of a triangle, the base angles of an isosceles triangle, medians that meet at a point, the parallelogram theorems, ASA, SAS and SSS, the angle-angle criterion, the tangent and the radius, and the area formula with sin C. Every formula and every number on a card was then tried a second way: areas against coordinates, volumes by adding thin slices, angle rules on plotted points and the coordinate rules on the point (3, 4). AAS, HL and SSA have one reference each and a worked check. Answers with a minus sign are stored with the keyboard minus, so a typed (x, -y) is marked right and (x, y) is not. Licence: own work

How it works

These geometry flash cards come in nine sets of twelve for grades 6 to 10: angle pairs, triangles, congruence, quadrilaterals, polygons, circles, area, volume and transformations. The page opens on angle pairs.

Read the full guide

Words on some cards, formulas on others

A word card gives the property and asks for the name: Two angles that add to 90° is complementary. A formula card names the figure, such as Volume of a cone, and the back gives ⅓πr²h. The line under every answer names the letters or works an example. Change sets under Show only.

One set, one sheet

Turn a card and the print links appear below it. A set is 12 cards, so it fills one sheet of Letter or A4: fronts on one side, backs on the other, flipped on the long edge. All 108 cards take nine sheets.

Trap card: two sides and an angle that is not between them (SSA) proves nothing. It can fit two different triangles.
In Quiz, under More options, write pi for π and ^ for a power: pi r^2. Keep the minus in (x, -y).
Pair up the Area cards: a triangle is half a parallelogram, and a kite is half the rectangle drawn round its diagonals.

Questions and answers

What grade are these geometry flash cards for?

Grades 6 to 10. The Common Core puts triangle and quadrilateral areas in grade 6, circles and angle pairs in grade 7, transformations and the Pythagorean theorem in grade 8, and congruence and circle theorems in high school. That is our reading, not an alignment.

What are the five triangle congruence rules?

SSS, SAS, ASA, AAS and HL. Each names the parts that must match: sides (S), angles (A), or the hypotenuse and a leg of two right triangles. SSA is not a rule, and three equal angles only prove that triangles are similar.

What is the difference between congruent and similar?

Congruent figures have the same shape and the same size. Similar figures have the same shape only: equal angles, and sides in one ratio, the scale factor. Congruent figures are similar with a scale factor of 1.