Every term used across the hub — in plain language first, with the formal definition alongside.
Angles between the two crossed lines, on opposite sides of the transversal — the "Z" pattern. When the two lines are parallel, they are equal.
Formal definition. A pair of interior angles lying on opposite sides of the transversal. If the two lines are parallel, alternate interior angles are congruent — and conversely.
The base angles of an isosceles triangle are the two angles that sit on the base — the angles opposite the two equal sides. They are always equal to each other.
Formal definition. In an isosceles triangle, the two angles opposite the congruent sides. The Base Angles Theorem states they are congruent, and its converse holds too: equal base angles force the opposite sides to be equal.
The fixed point a dilation grows or shrinks a figure from. Every point slides straight toward it or straight away from it along a ray, and the center itself is the one point that never moves.
Formal definition. The unique fixed point of a dilation. For a center and scale factor , each point maps to the point on ray with .
The pin a turn spins around — that one point stays put while everything else travels around it.
Formal definition. The fixed point about which a rotation turns the plane; every point and its image lie on a circle centered there, so each point stays the same distance from the center.
Two shapes are congruent when one fits exactly on top of the other — same size, same shape, even if it was slid, flipped, or turned to get there.
Formal definition. Two figures are congruent if and only if there is a sequence of rigid motions (translations, reflections, rotations) that maps one figure onto the other.
Angles in matching corners at the two crossings a transversal makes — same side of the transversal, same position relative to each crossed line. When the two crossed lines are parallel, corresponding angles are equal.
Formal definition. A pair of angles, one at each intersection of a transversal with two lines, lying on the same side of the transversal and in the same position relative to each crossed line. If the two lines are parallel, corresponding angles are congruent — and conversely, equal corresponding angles force the lines to be parallel.
CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. Once you know two triangles are congruent, every matching pair of sides and every matching pair of angles is congruent too.
How to use it. Read the partners straight from the congruence statement, pairing the letters in the same position: means , , , so for example and .
An exterior angle of a triangle is the angle made when you extend one side past a vertex — it sits right next to an interior angle and together they make a straight line.
Formal definition. The angle between one side of a triangle and the extension of an adjacent side. It is supplementary to the interior angle at that vertex, and equals the sum of the two remote (non-adjacent) interior angles.
The shape you end up with, after a transformation has moved, flipped, turned, or resized the original.
Formal definition. The figure produced by applying a transformation; if a transformation maps point to point , then is the image of , and is usually named with a prime mark.
An isosceles triangle has (at least) two sides of equal length. The two equal sides are its legs, the third side is the base, and the two angles at the base are always equal to each other.
Formal definition. A triangle with at least two congruent sides. The angles opposite those sides (the base angles) are congruent.
The mirror line of a flip: the original shape and its image sit on opposite sides of it, the same distance away.
Formal definition. The line over which a reflection maps each point to its image; it is the perpendicular bisector of every segment joining a point to its image.
A line you could fold a shape across so the two halves land exactly on top of each other. A square has four; a scalene triangle has none.
Formal definition. A line is a line of symmetry of a figure when reflecting the figure over maps it exactly onto itself.
Two angles that sit side by side along a straight line. Their outer sides point in exactly opposite directions, so together they make a straight angle and add to 180°.
Formal definition. Two adjacent angles whose non-common sides form a straight line (opposite rays). The angles in a linear pair are supplementary.
A midsegment of a triangle is the segment joining the midpoints of two of its sides. It always runs parallel to the third side and is exactly half as long.
Formal definition. The segment connecting the midpoints of two sides of a triangle. It is parallel to the third side and half its length.
The shape you start with, before a transformation moves, flips, turns, or resizes it.
Formal definition. The original figure to which a transformation is applied; if a transformation maps point to point , then is the preimage of .
For a given exterior angle of a triangle, the remote interior angles are the two interior angles that are not next to it — the two angles "across" the triangle from where the side was extended.
Formal definition. The two interior angles of a triangle that are not adjacent to a given exterior angle. Their measures sum to that exterior angle.
A move that doesn't stretch, shrink, or bend the shape — it slides, flips, or turns as one solid piece, like moving a paper cutout.
Formal definition. A transformation of the plane that preserves distance and angle measure. Translations, reflections, rotations, and any sequence of them are rigid motions.
A shape has rotation symmetry when you can spin it about its center by less than a full turn and it lands exactly on itself. A square has it (quarter turns); a shape with three unequal sides does not.
Formal definition. A figure has rotation symmetry when some rotation about its center, by an angle strictly between and , maps the figure onto itself. For a figure with -fold symmetry the smallest such angle is .
Angles between the two crossed lines, on the same side of the transversal — the "C" pattern. When the two lines are parallel, they add to 180°. Also called consecutive interior or co-interior angles.
Formal definition. A pair of interior angles lying on the same side of the transversal. If the two lines are parallel, same-side interior angles are supplementary — and conversely.
Two shapes are similar when they have the same shape but not necessarily the same size — one is a scaled copy of the other. All the angles match, and the side lengths stay in proportion.
Formal definition. Two figures are similar if and only if some sequence of rigid motions and dilations maps one onto the other. Equivalently, their corresponding angles are equal and their corresponding side lengths share a common ratio — the scale factor.
Two angles are supplementary when their measures add up to 180° — a straight line's worth of turn. They don't have to be next to each other.
Formal definition. Two angles whose measures sum to 180°.
A line that cuts across two or more other lines. Each crossing it makes has its own set of angles.
Formal definition. A line that intersects two or more coplanar lines at distinct points.
The triangle inequality is the rule that decides whether three lengths can form a triangle at all: any two sides, added together, must be longer than the third. If they aren't, the short sides can't reach across to meet.
Formal definition. For side lengths , , of a triangle, , , and . Three lengths form a triangle exactly when all three hold; equality gives a degenerate (flat) triangle.
When two lines cross, the two angles sitting directly across from each other — sharing only the crossing point — are vertical angles. They are always equal.
Formal definition. A pair of non-adjacent angles formed by two intersecting lines; they share a vertex but no side. Vertical angles are always congruent.