Praxis Geometry key terms

The basic building blocks of geometry-point, line, and plane-that are accepted without formal definition and used to define all other figures.

An exact location in space with no size, length, or width, usually named with a capital letter and drawn as a dot.

A straight one-dimensional figure extending infinitely in both directions, determined by any two distinct points on it.

A flat two-dimensional surface extending infinitely in all directions, determined by any three noncollinear points.

The part of a line consisting of two endpoints and every point between them; it has a measurable finite length.

The part of a line starting at one endpoint and extending infinitely in one direction, named with the endpoint first.

Three or more points that all lie on a single straight line; any two points are always collinear.

Points that all lie in the same plane; unlike collinearity, coplanarity can hold for any number of points.

The point on a segment that divides it into two congruent halves, located exactly halfway between the endpoints.

A figure formed by two rays sharing a common endpoint called the vertex, measured in degrees or radians.

Two angles whose measures add to exactly 90 degrees; each angle is called the complement of the other.

Two angles whose measures add to exactly 180 degrees; each angle is called the supplement of the other.

Two adjacent angles formed by opposite rays; they share a vertex and side and are always supplementary.

Two nonadjacent angles formed by two intersecting lines; they are congruent and have equal measures.

A line intersecting two or more other lines, creating angle pairs such as alternate interior and corresponding angles.

A pair of angles on opposite sides of a transversal and between the two intersected lines; congruent when those lines are parallel.

Angles in matching positions at each intersection made by a transversal; congruent when the intersected lines are parallel.

Coplanar lines that never intersect, no matter how far extended; they have equal slopes in the coordinate plane.

Two lines that intersect to form right angles; their slopes are negative reciprocals in the coordinate plane.

A line, ray, or segment perpendicular to a segment at its midpoint; every point on it is equidistant from the segment's endpoints.

A ray dividing an angle into two congruent angles, with every point on it equidistant from the sides of the angle.

Rules-SSS, SAS, ASA, AAS, and HL-that guarantee two triangles are congruent from limited matching side and angle information.

An abbreviation meaning corresponding parts of congruent triangles are congruent, used to justify statements after congruence is proven.

Triangles with congruent corresponding angles and proportional corresponding sides; AA, SSS, and SAS similarity establish this relationship.

The ratio comparing corresponding lengths of a figure and its image under a dilation or between similar figures.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a squared plus b squared equals c squared.

Right triangles with predictable side ratios: the 45-45-90 triangle has legs in ratio 1 to 1 to the square root of 2; the 30-60-90 triangle has ratio 1 to square root of 3 to 2.

Ratios of side lengths in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

In any triangle, the ratio of a side to the sine of its opposite angle is constant: a over sine A equals b over sine B equals c over sine C.

A generalization of the Pythagorean theorem for any triangle: c squared equals a squared plus b squared minus 2ab cosine C.

The interior angles of an n-sided polygon sum to (n minus 2) times 180 degrees; exterior angles of any convex polygon sum to 360 degrees.

An equilateral and equiangular polygon; its interior angle measures (n minus 2) times 180 divided by n degrees.

A quadrilateral with both pairs of opposite sides parallel; opposite sides and angles are congruent and diagonals bisect each other.

A parallelogram with four congruent sides; its diagonals are perpendicular and bisect its angles.

Relationships among central, inscribed, and tangent angles: an inscribed angle is half its intercepted arc, and a tangent is perpendicular to the radius at the point of tangency.

An angle whose vertex lies on a circle and whose sides are chords; its measure is half that of its intercepted arc.

An angle whose vertex is the center of a circle; its measure equals the measure of the arc it intercepts.

The distance along a circle between two points, computed as the central angle in degrees divided by 360, times the circumference.

The region of a circle bounded by two radii and an arc, computed as the central angle divided by 360, times pi times the radius squared.

A line intersecting a circle at exactly one point, called the point of tangency, and perpendicular to the radius drawn there.

Expressions giving space occupied by solids: prisms and cylinders use base area times height, pyramids and cones use one third of that, and spheres use four thirds pi r cubed.

The total area of all faces or curved surfaces of a solid, measured in square units, distinct from its volume.

A transformation that enlarges or reduces a figure about a center point by a scale factor, preserving shape but not size.

A transformation-translation, rotation, or reflection-that preserves distance and angle measure, producing a congruent image.

A composition of a reflection and a translation parallel to the line of reflection; it is an isometry with no fixed points.

The expression giving the distance between two coordinate points: the square root of the sum of the squared differences of their x- and y-coordinates.

The rule giving a segment's midpoint coordinates as the averages of the endpoints' x-coordinates and of their y-coordinates.

A proof that places a figure on the coordinate plane, assigns general coordinates, and uses slope, distance, and midpoint formulas to verify a claim.

A statement in if-then form with a hypothesis and conclusion; its converse swaps those parts and is not automatically true.

The statement formed by negating and swapping the hypothesis and conclusion of a conditional; it is logically equivalent to the original.

A proof technique that assumes the negation of the desired conclusion and derives an impossibility, thereby establishing the original claim.