Geometry & Figures

Unknown angles fall out of a small set of facts: a straight line is 180 degrees, a triangle sums to 180, a quadrilateral to 360, angles combine and split, and parallel lines give equal corresponding/alternate angles. Learners apply these from clock faces and single figures up to polygon angle sums via triangulation and auxiliary lines. The method is choosing the right fact and chaining equations.

Equal sides force equal base angles, so spotting isosceles or equilateral triangles (often from equal radii or a diagonal) lets you chain known angles to an unknown one. Learners identify equal-side triangles inside circles and polygons and step through the angle relations. It fuses side-equality with angle-sum facts.

Flips, turns, and folds move a figure without changing its side lengths or angle measures, and repeated transformations can return it to start (two flips, four quarter-turns) or be undone in reverse. Learners reason about flip/turn identities, rotation centers, and folded figures whose matched parts are congruent. The invariance principle underlies all of congruence.

Special quadrilaterals are characterized by their diagonals: a parallelogram's diagonals bisect each other, a square's are equal and perpendicular bisectors. Learners use these defining properties to find lengths and angles. It sharpens classification of quadrilaterals by structural property rather than appearance.

Every radius of a circle is equal and the diameter is twice the radius; chaining these through touching or nested circles turns center-to-center segments and polygon sides into sums of radii. Learners fit circles in rectangles, build polygons joining centers, and compute perimeters mixing arcs and straight radius parts. It is geometric reasoning by equal-length substitution.

Small pieces combine into larger ones, so counting all triangles, quadrilaterals, segments, or vertices in a composite figure requires an organized sweep by size or type to avoid missing or double-counting. Learners count hidden shapes in grids and composites and tally dots, lines, and faces. The discipline is exhaustive, non-overlapping enumeration.

Reason about a rectangular prism, cube, or die through its net, faces, edges, and vertices: which faces are opposite or adjacent, hidden versus visible edges, folding a net, and positional face counts.

How much skin a solid has and how much space it fills. Surface area is every outward face added once, so faces that touch stop counting; volume is a base stacked to a height, so cutting a piece away removes exactly that piece. Both run backwards too: a known area or volume gives back a missing edge.

A pile of unit cubes described by what it looks like from above, in front and from the side. Each view is a projection that keeps only the tallest cube in its line, so views constrain the pile without fixing it -- some positions are pinned down and others are still free, which is what makes counting all the possible piles a real question.

Each linear equation in two unknowns is a line, and a solution of the system is where the lines cross. So 'no solution' is parallel lines, 'infinitely many' is the same line, an equation saying a product of two linear expressions is zero is the union of two lines, and three distinct lines fail to make a triangle exactly when two are parallel or all three pass through one point.

A line through a fixed point turns like a hand on a dial. When the whole segment lies on one side of the vertical line through the fixed point, the slope to a point of the segment runs from one endpoint's slope to the other's, so 'meets the segment' is the slope range between the endpoints, ends included. Halving an area works the same way: decide which side the line must cross, then fix the crossing point by area. A line through the centre of a centrally symmetric figure always halves it.