Regents Examination in Geometry: Glossary of Geometry Terms

A

Acute angle – An angle whose measure is greater than 0° and less than 90°.
Acute triangle – A triangle whose angles are all acute angles.
Adjacent angles – Two angles that share a common vertex and one side but do not share interior points.
Altitude – A segment from a vertex of a triangle perpendicular to the opposite side.
Angle – A figure formed by two rays with a common endpoint. Symbol is ∠.
Angle bisector – A line, segment, or ray that divides an angle into two congruent angles.
Angle measure – The amount of opening of an angle, measured in degrees or radians.
Angle of depression – The angle formed by the horizontal and the line of sight when looking downward to an object.
Angle of elevation – The angle formed by the horizontal and the line of sight when looking upward to an object.
Angle of rotation – The angle measure by which a figure or point spins around a center point.
Arc – A portion of a circle with two endpoints on the circle. A major arc measures more than 180°; a minor arc measures less than 180°; a semicircle measures 180°. Symbol is .
Arc length – The distance between the endpoints of an arc. Arc length equals rθ where θ is the arc measure in radians and r is the radius.
Arc measure – The angle measure of an arc, equal to the measure of the central angle that intercepts the arc.
Axis of symmetry – A line that divides a figure into two congruent parts.


B

Base (of a circular cone) – The circular face of a cone; it is opposite the apex.
Base (of an isosceles triangle) – The non-congruent side of an isosceles triangle.
Base (of a pyramid) – The polygonal face of a pyramid that is opposite the apex.
Bases (of a prism) – A pair of faces of a prism that are parallel, congruent polygons.
Bases (of a circular cylinder) – The pair of congruent parallel circular faces of a cylinder.
Base Angles – The angles formed at each end of the base of an isosceles triangle.
Bisector (Segment bisector) – A line, segment, or ray that passes through the midpoint of a segment.


C

Cavalieri’s principle – If two solids are contained between two parallel planes and every parallel plane between these two planes intercepts regions of equal area, then the solids have equal volume. Also any two parallel planes intercept two solids of equal volume.
Center of a regular polygon – The center of the inscribed or circumscribed circle of a regular polygon.
Center of dilation – The fixed point reference point used to determine the expansion or contraction of lengths in a dilation. The center of dilation is the only invariant point in a dilation.
Center–radius equation of a circle – A circle can be represented on the coordinate plane by (x – h)2 + (y – k)2 = r2 , where the point (h, k) is the center of the circle and r is the radius.
Central angle – An angle in a circle formed by two distinct radii.
Centroid of a triangle – The point of concurrency of the three medians of a triangle.
Chord – A segment whose endpoints lie on a circle.
Circle – The set of points that are equidistant from a fixed center point. Symbol is .
Circumcenter – The point that is the center of the circle circumscribed about a polygon, equidistant from the vertices of a polygon, and the point of concurrency of the perpendicular bisectors of a triangle.
Circumference – The distance around a circle. Circumference = 2 π ·radius.
Coincide (coincident) – Figures that lie entirely on each other.
Collinear – Points that lie on the same line.
Compass – A tool for drawing accurate circles and arcs.
Complementary angles – Angles whose measures sum to 90°.
Completing the square – A method used to rewrite a quadratic expression of the form ax2 + bx + c as a squared binomial of the form (dx + e)2.
Composition of transformations – A sequence of transformations in a specified order. Notation is transformation 2 ◦ transformation 1. The transformation on the right is performed first.
Concave polygon – A polygon with at least one diagonal outside the polygon.
Concentric circles – Circles with the same center.
Concurrent – When three or more lines all intersect at a single point.
Cone (circular) – A solid figure with a single circular base and a curved lateral face that tapers to a point called the apex.
Congruent – Figures with the same size and shape. Symbol is ≅ .
Construction – A figure drawn with only a compass and straightedge.
Convex polygon – A polygon whose diagonals all lie within the polygon.
Coplanar – Figures that lie in the same plane.
Corresponding parts – A pair of parts (usually points, sides, or angles) of two figures that are paired together through a specified relationship, such as a congruence or similarity statement or a transformation function.
Cosine of an angle – In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.
CPCTC – Corresponding parts of congruent triangles are congruent.
Cross section – The intersection of a plane with a solid.
Cube – A prism whose faces are all squares.
Cubic unit – The amount of space occupied by a cube 1 unit of length in each dimension. (1 ft3 is the amount of space occupied by a 1 ft × 1 ft × 1 ft cube).
Cylinder (circular) – A solid with two parallel, congruent circular bases and a curved lateral face.


D

Degree – A unit of angle measure equal to  of a complete rotation.
Diagonal – A segment in a polygon whose endpoints are nonconsecutive vertices of the polygon.
Diameter – A chord that passes through the center of a circle.
Dilation – A similarity transformation about a center point O that maps preimage P to image Pʹ such that O , P , and Pʹ are collinear and OPʹ = k · OP. Notation is Dk.
Direct transformation – A transformation that preserves orientation.
Distance from a point to a line – The length of the segment from the point perpendicular to the line.


E

Edge of a solid – The intersection of two polygonal faces of a solid.
Endpoint – The two points that bound a segment.
Equiangular – A figure whose angles all have the same measure.
Equidistant – The same distance from two or more points.
Equilateral – A figure whose sides all have the same length.
Exterior angle of a polygon – The angle formed by a side of a polygon and the extension of an adjacent side.


F

Face of a solid – Any of the surfaces that bound a solid.
Figure – Any set of points. The points may form a segment, line, ray, plane, polygon, curve, solid, and so on.


G

Geometric mean – See Mean proportional .
Great circle – The largest circle that can be drawn on a sphere. The circle formed by the intersection of a sphere and a plane passing through the center of the sphere.


H

Hexagon – A 6-sided polygon.
Hypotenuse – The longest side of a right triangle. It is always opposite the right angle.


I

Identity transformation – A transformation in which the preimage and image are coincident.
Image – The figure that results from applying a transformation to an initial figure called the preimage.
Incenter of a triangle – The point that is the center of the inscribed circle of a triangle, equidistant from the 3 sides of a triangle, and the point of concurrency of the three angle bisectors of a triangle.
Inscribed angle – An angle in a circle formed by two chords with a common endpoint on the circle.
Inscribed circle of a triangle – A circle that is tangent to all three sides of the triangle.
Intersecting – Figures that share at least one common point.
Isometry – See Rigid motion.
Isosceles triangle – A triangle with two congruent sides.
Isosceles trapezoid – A trapezoid with congruent legs.


K

Kite – A polygon with two distinct pairs of adjacent congruent sides. The opposite sides are not congruent.


L

Lateral edge – The intersection between two lateral faces of a polyhedron.
Lateral face – Any face of a polyhedron that is not a base.
Line – One of the undefined terms in geometry. An infinitely long set of points that has no width or thickness. Symbol is ↔.
Linear pair of angles – Two adjacent supplementary angles.
Line symmetry – A line over which one half of a figure can be reflected and mapped onto the other half of the figure.
Line reflection – A rigid motion in which every point P is transformed to a point Pʹ such that the line is the perpendicular bisector of
.


M

Major arc – An arc whose degree measure is greater that 180°.
Map (Mapping) – A pairing of every point in a preimage with a point in an image. Reflections, rotations, translations, and dilations are oneto-one mappings because every point in the preimage maps to exactly one point in the image and every point in the image maps to exactly one point in the preimage.
Mean proportional (Geometric mean) – The geometric mean of two numbers, a and b , is the square root of their product. If
 then m is the geometric mean.
Median of a triangle – The segment from a vertex of a triangle to the midpoint of the opposite side.
Midpoint – A point that divides a segment into two congruent segments.
Midsegment of a triangle – A segment joining the midpoints of two sides of a triangle.
Midsegment (median) of a trapezoid – A segment joining the midpoints of the two legs of a trapezoid.
Minor arc – An arc whose degree measure is less than 180°.


N

Noncollinear – Points that do not lie on the same line.
Noncoplanar – Points or lines that do not lie on the same plane.


O

Obtuse angle – An angle whose measure is greater than 90° and less than 180°.
Octagon – A polygon with 8 sides.
Opposite rays – Two rays with a common endpoint that together form a straight line.
Opposite transformation – A transformation that changes the orientation of a figure.
Orientation – The order in which the points of a figure are encountered when moving around a figure. The orientation can be clockwise or counterclockwise.
Orthocenter – The point of concurrence of the three altitudes of a triangle.


P

Parallel lines – Coplanar lines that do not intersect. Symbol is ||.
Parallelogram – A quadrilateral with two pairs of opposite parallel sides.
Parallel planes – Planes that do not intersect.
Pentagon – A polygon with 5 sides.
Perimeter – The sum of the lengths of the sides of a polygon.
Perpendicular – Intersecting at right angles. Symbol is ⊥.
Perpendicular bisector – A line, segment, or ray perpendicular to another segment at its midpoint.
Pi – The ratio of the circumference of a circle to its diameter. Pi is an irrational number whose value is approximately 3.14159. Symbol is π.
Plane – One of the undefined terms in geometry. A set of points with no thickness that extends infinitely in two directions. It is usually visualized as a flat surface.
Point – An undefined term in geometry. A location in space with no length, width, or thickness. Symbol is •.
Point of tangency – The point where a tangent intersects a curve.
Point reflection – A transformation in which a specified center point is the midpoint of each of the segments connecting a point in the preimage with a corresponding point in the image. It is equivalent to a rotation of 180°.
Point-slope equation of a line – A form of the equation of a line, y –y1 = m(x – x1), where m is the slope and (x1, y1) are the coordinates of a point on the line.
Point symmetry – A figure has point symmetry if every point in the figure is a 180° rotation of a corresponding point in the same figure.
Polygon – A closed planar figure whose sides are segments that intersect only at their endpoints (do not overlap).
Polyhedron – Plural: polyhedra. A solid figure in which each face is a polygon. Prisms and pyramids are examples of polyhedra.
Postulate – A statement that is accepted to be true without proof.
Preimage – The original figure that is acted on by a transformation.
Preserves – Remains unchanged.
Prism – A polyhedron with two congruent, parallel polygons for bases and whose lateral faces are parallelograms.
Proportion – An equation that states two ratios are equal. For example, . The cross products of a proportion are equal, a·d = b·c.
Pyramid – A polyhedron with one polygonal base and triangles for lateral faces.
Pythagorean theorem – In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse, or a2 + b2 = c2, where a and b are the lengths of the two legs and c is the length of the hypotenuse.


Q

Quadratic equation – An equation in the form ax2 + bx + c = 0, where a ≠ 0.
Quadratic formula – A formula for finding the two solutions to a quadratic equation of the form ax2 + bx + c = 0, .
Quadrilateral – A polygon with four sides.


R

Radian – An angle measure in which one full rotation is 2π radians. Also 1 radian is the measure of an arc such that the arc’s length is equal to the radius of that circle.
Radius – A segment from the center of a circle to a point on the circle.
Ray – A portion of a line starting at an endpoint and including all points on one side of the endpoint. Symbol is →.
Rectangle – A parallelogram with right angles.
Reflection – See Line reflection and Point reflection.
Reflexive property of equality – Any quantity is equal to itself. For figures, any figure is congruent to itself.
Regular polygon – A polygon in which all sides are congruent and all angles are congruent.
Rhombus – A parallelogram with four congruent sides.
Right angle – An angle that measures 90°.
Right circular cone – A cone with a circular base and whose altitude passes through the center of the base.
Right circular cylinder – A cylinder with circular bases and whose altitude passes through the center of the bases.
Right pyramid – A pyramid whose faces are isosceles triangles.
Right triangle – A triangle that contains a right angle.
Rigid motion (Isometry) – A transformation that preserves distance. The image and preimage are congruent under a rigid motion. Translations, reflections, and rotations are isometries.
Rotation – A rigid motion in which every point, P , in the preimage spins by a fixed angle around a center point, C , to a point Pʹ. The distance to the center point is preserved.
Rotational symmetry – A figure has rotational symmetry if it maps onto itself after a rotation of less than 360°.


S

Scale drawing – A drawing of a figure or an object that represents a dilation of the actual figure or object. A drawing of an object in which every length is enlarged or reduced by the same scale factor.
Scale factor – The ratio by which a figure is enlarged or reduced by a dilation.
Scalene triangle – A triangle in which no sides have the same length.
Secant of a circle – A line that intersects a circle in exactly two points.
Sector of a circle – A region bounded by a central angle of a circle and the arc it intersects.
Segment – A portion of a line bounded by two endpoints.
Segment bisector – See Bisector.
Semicircle – An arc that measures 180°.
Similar polygons – Two polygons with the same shape but not necessarily the same size.
Similarity transformation – A transformation in which the preimage and the image are similar. A transformation that includes a dilation.
Sine of an angle – In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.
Skew lines – Two lines that are not coplanar.
Slant height – The distance along a lateral face of a solid from the apex perpendicular to the opposite edge.
Slope – A numerical measure of the steepness of a line. In the coordinate plane, the slope of a line equals the change in the y-coordinates divided by the change in the x-coordinates between any two points. The slope of a vertical line is undefined.
Slope-intercept equation of a line – A form of the equation of a line, y = mx + b, where m is the slope and b is the y-intercept.
Solid figure – 3-dimensional figure fully enclosed by surfaces.
Sphere – A solid comprised of the set of points in space that are a fixed distance from a center point.
Square – A parallelogram with four right angles and four congruent sides.
Straightedge – A ruler with no length marking, used for constructing straight lines.
Substitution property of equality – The property that a quantity can be replaced by an equal quantity in an equation.
Subtraction property of equality – If the same or equal quantities are subtracted from the same or equal quantities, the differences are equal.
Supplementary angles – Two angles whose measures add to 180°.
Surface area – The sum of the areas of all the faces or curved surfaces of a solid figure.


T

Tangent of an angle – In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the side adjacent to the angle.
Tangent to a circle – A line, coplanar with a circle, that intersects the circle at only one point.
Theorem – A general statement that can be proved.
Transformation – A one-to-one function that maps a set of points, called the preimage, to a new set of points, called the image.
Transitive property of congruence – The property that states if figure A ≅ figure B and figure B ≅ figure C , then figure A ≅ figure C.
Transitive property of equality – The property that states if a = b and b = c , then a = c.
Translation – A rigid motion that slides every point in the preimage in the same direction and by the same distance.
Transversal – A line that intersects two or more other lines at different points.
Trapezoid – A quadrilateral that has one and only one pair of parallel sides.
Truth value – The value of true or false that is assigned to a statement.


U

Undefined terms – Terms that cannot be formally defined using previously defined terms. In geometry, these traditionally include point, line, and plane.


V

Vector – A quantity that has both magnitude and direction; represented geometrically by a directed line segment. Symbol is →.
Vertex – Plural: vertices. The point of intersection of two consecutive sides of a polygon.
Vertex angle – The angles formed by the two congruent sides in an isosceles triangle.
Vertical angles – The pairs of opposite angles formed by two intersecting lines.
Volume – The amount of space occupied by a solid figure measured in cubic units (in.3 , cm3 , and so on). The number of nonoverlapping unit cubes that can fit into the interior of a solid.


Z

Zero product property – The property that states if a · b = 0, then a = 0, or b = 0, or a and b = 0.