Geometry DictionarySasha Vasserman

Two triangles are similar if two pairs of corresponding angles are congruent

AA Theorem of Similarity

Two triangles are congruent if two pairs of corresponding angles are congruent and a non-included pair of corresponding sides are congruent

AAS Theorem of Congruence

The x-coordinate of a point in the coordinate plane

Abscissa

For a number x, denoted by |x |, its distance from 0 on the number line. Thus, |x| always represents a nonnegative number

Absolute value

An angle whose degree measure is < 0 and > 90

Acute angle

A triangle with three acute angles

Acute traingle

Two angles that have the same vertex and share one side, but do not have any inferior points in common

Adjacent angles

Pairs of angles formed when a transversal intersects two lines. The two angles ion each pair are between the two lines, have different vertices, and lie on opposite sides of the transversal

Alternate interior angles

A segment that is perpendicular to the side of the figure to which it is drawn

Altitude

The union of two rays that have the same end point

Angle

A line or any part of a line that contains the vertex of an angle and that divides the angle into two congruent angles. An angle has exactly one angle bisector

Angle bisector

An angle formed by a horizontal ray of sight and the ray that is the line of sight to an object below the horizontal ray

Angle of depression

An angle formed by a horizontal ray of sight and the ray that is the line of sight to an object above the horizontal ray

Angle of elevation

For a regular polygon, the radius of its inscribed circle

Apothem

The minor arc of a circle whose end points are the end points of a chord. If the chord is a diameter, then either semicircle is an arc of the diameter

Arc of a chord

For a plane geometric figure, the number of square units it contains

Area

Two triangles are congruent if two pairs of corresponding angles are congruent and the sides included by these angles are congruent

ASA Theorem of Congruence

The congruent angles that lie opposite the congruent sides of an isosceles triangle

Base angles of an isosceles triangle

The non-congruent side of the isosceles triangle

Base of an isosceles triangle

The parallel sides of a trapezoid

Base of a trapezoid

A term that refers to the order of three collinear points. If A, B, and C are three different collinear points, point C us between points A and b if AC + CB = AB

Betweenness of points

To divide into two equal parts

Bisect

The common center of the circles inscribed and circumscribed in the polygon

Center of a regular polygon

An angle whose vertex is at the center of a circle, and whose sides are radii

Central angle of a circle

An angle whose vertex is the center if the regular polygon and whose sides terminate at consecutive vertices of the polygon

Central angle of a regular polygon

The point at which three medians of the triangle intersect

Centroid of a triangle

A segment whose end points are on the circle

Chord of a circle

The set of all points in a plane at a fixed distance from a given point called the center. The fixed distance is called the radius of the circle. An equation of a circle with center at point (h, k) and radius length r is (x - h²) + (y- k)² = r²

Circle

The distance around a circle

Circumference of a circle

A circle that passes through each vertex of the polygon

Circumscribed circle about a polygon

A polygon that has all of its sides tangent to the circle

Circumscribed polygon about a circle

Points that lie on the same line

Collinear points

A line that is tangent to both circles, and does not intersect the line segment whose end points are the centers of the two circles

Common external tangent to two circles

A line that is tangent to both circles, and intersects the line segment whose end points are the centers of the two circles

Common internal tangent to two circles

Two angles whose measures add up to 90°

Complementary angles

A sequence of two or more transformations in which each transformation after the first is preformed on the image of the transformation that was applied before it

Composite transformation

Circles in the same plane that have the same center but have radii of different lengths

Concentric circles

Angles that have the same measure

Congruent angles

Circles with congruent radii

Congruent circles

Line segments that have the same length

Congruent line segments

Polygons with the same number of sides that have the same size and same shape. The symbol for congruence is ≅

Congruent polygons

Triangles whose vertices can be paired so that any one on the following conditions is true: (1) the sides of one triangle are congruent to the corresponding sides of the other triangle (SSS ≅ SSS); (2) two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle (SAS ≅ SAS); (3) two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle (ASA ≅ ASA); (4) two angles and the side opposite one of these angles of one triangle are congruent to the corresponding parts of the other triangle ( AAS ≅ AAS). Two right triangles are congruent if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts the other triangle (Hy – Leg ≅ Hy – leg)

Congruent triangles

Another conditional statement formed by interchanging the hypothesis (“Given”) with the conclusion (“To Prove”) of the original statement

Converse of a conditional statement

A polygon each of whose interior angles measures less than 180°

Convex polygon

A plane that is divided into four equal regions, called quadrants, by a horizontal number line and a vertical number line, called axes, intersecting at their zero points, called the origin. Each point in a coordinate plane is located by an ordered pair of numbers of the form (x, y). The first member, x, of the ordered pair gives the directed distance of the zero point of the x-axis (horizontal). The second member, y, of the ordered pair gives the directed distance of the point from the zero point of the y-axis

Coordinate plane

A theorem that can easily be proved by means of a closely related theorem

Corollary

Pairs of angles formed when a transversal intersects two lines. The two angles in each pair lie on the same side of the transversal, but one angle is between the two lines, and the other is exterior to the two lines

Corresponding angle

The ratio of the length of the leg that is adjacent to the acute angle to the length of the hypothenuse

Cosine of an acute angle of a right triangle

A polygon with 10 sides

Decagon

A step-by-step process by which a set of accepted facts is used to arrive at a conclusion

Deductive reasoning

A unit of angle measure. One degree is the measure of an angle formed by 1/360 of one complete rotation of a ray about its end point

Degree

A line segment whose end points are nonconsecutive vertices of the polygon

Diagonal of a polygon

A chord of the circle that contains the center of the circle

Diameter of a circle

A size transformation that produces an image similar to the original figure

Dilation

An isometry that pressserves orientation

Direct isometry

A formula used to find the length of the segment determined by two points in the coordinate plane. The distance d between two points , A and B, is given by the formula d = the square root of ((X of point B – X of point A)squared + (Y of point B – Y of point A) squared)

Distance Formula

The length of the perpendicular segment from the point to the line

Distance form a point to a line

A polygon with 12 sides

Dodecagon

A polygon in which all the angles have the same measure

Equiangular Polygon

A triangle in which all three angles have the same measure

Equiangular Triangle

Having the same distance

Equidistant

A polygon in which all the sides have the same length

Equilateral Polygon

A triangle whose three sides have the same length

Equilateral Triangle

An angle formed by a side of the polygon and the extension of an adjacent side of the polygon

Exterior angle of a polygon

Tangent circles that lie on opposite sides of the common tangent

Externally tangent circles

The first and fourth terms in a proportion. In the proportion a over b = c over d, a and d are the extremes

Extremes

The composition of a reflection in a line and a translation in the direction parallel to the reflecting line

Glide reflection

The set of points in a plane that lie on one side of a line

Half-plane

A polygon with six sides

Hexagon

The side of a right triangle that is opposite to the right angle

Hypotenuse

The point at which three bisectors of the triangle intersect

Incenter of a triangle

A method of proof in which each possibility except the one that needs to be proved is eliminated by showing that it contradicts some known or given fact

Indirect proof

An angle whose vertex lies on the circle and whose sides are chords of the circle

Inscribed angle of a circle

A circle that is tangent to each side of the polygon

Inscribed circle of a polygon

A polygon that has all of its vertices on a circle

Inscribed polygon

Tangent circles that lie on one side of the common tangent

Internally tangent circles

A transformation that produces an image congruent to the original figure

Isometry

A trapezoid whose nonparallel sides called legs, have the same length

Isosceles trapezoid

A triangle with two sides, called legs that have the same length

Isosceles triangle

Either of the two sides of the right triangle that are not opposite the right angle

Leg of a right triangle

A term undefined in geometry; a line can be described as a continuous set of points forming a straight path that extends indefinitely in two opposite directions

Line

The line segment whose end points are the centers of the circles

Line of centers of two circles

Part of a line that consists of two different points on a line called end points, and the set of all points on the line that ate between them. AB refers to the distance of a line segment with end points A and B, whereas AB with a line of it refers to the segment itself

Line segment

When a line can be drawn that divides the figure into two parts that coincide when folded along the line

Line symmetery

The set of all points, and only those points, that satisfy a given condition

Locus (Plural: Loci )

An arc of a circle whose degree measure is greater than 180°

Major arc

The two middle terms of a proportion. In the proportion a over b = c over d, then either b or c are called the mean proportional between a and d

Means

A line segment whose end points are the midpoints of the legs of the trapezoid

Median of a trapezoid

A line segment whose end points are a vertex of the triangle and the midpoint of the side opposite that vertex

Median of a triangle

A formula used to find the coordinates of the midpoint of a line segment in the coordinate plane. The midpoint of a line segment whose points are A(x, y) and B(X, Y) is ( (x + X over 2) + ( y + Y) over 2))

Midpoint formula

The point on a line segment that divides the segment into two segments that have the same length

Midpoint of a line segment

An arc of a circle whose degree measure is less than 180°

Minor arc

An angle whose degree measure is greater than 90° and less than 180°

Obtuse angle

A triangle that contains an obtuse angle

Obtuse triangle

A polygon with eight sides

Octagon

An isometry that reverses orientation

Opposite isometry

Two rays that have the same end point and form a line

Opposite rays

The y-coordinate of a point in the coordinate plane

Ordinate

The zero point on a number line

Origin

The point at which the three altitudes of the triangle intersect

Orthocenter of a triangle

Lines in the same plane that do not intersect

Parallel lines

A quadrilateral that has two pairs of parallel sides

Parallelogram

A polygon with five sides

Pentagon

The sum of the lengths of the sides of the polygon

Perimeter of a polygon

A line, ray, or line segment that is perpendicular to the segment at its midpoint

Perpendicular bisector of a segment

Two lines that intersect at 90° angles

Perpendicular lines

A term undefined in geometry; a plane can be described as a flat surface that extends indefinitely in all directions

Plane

A term undefined in geometry; a point can be described as a dot with no size that indicates location

Point

A figure with 180° rotational symmetry

Point symmetry

A closed figure in a plane whose sides are line segments that intersect at their end points

Polygon

A statement whose truth is accepted without proof

Postulate

An equation that states that two ratios are equal. In a proportion, the product of the means equals the product of the extremes

Proportion

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse

Pythagorean theorem

One of the four equal rectangular regions into which the coordinate plane is divided

Quadrant of a coordinate plane

A polygon with four sides

Quadrilateral

A line segment whose end points are the center of the circle and any point on a circle

Radius (plural: Radii ) of a circle

The radius of its circumscribed circle

Radius of a regular polygon

A comparison of two numbers by division. The ratio of a to b can be represented by the fraction a over b, provided that b is not equal to zero

Ratio

The constant ratio of the lengths of any two corresponding sides

Ratio of similitude of two similar polygons

The part of a line that consists of a fixed point, called an end point, and the set of all points on one side of the end point

Ray

The number that, when multiplied by the original number, gives 1. For example, the reciprocal over 1 over 5 is 5 over 1, AKA, 5. 5 over 1 times 1 over 5 equals 1

Reciprocal of a nonzero number

A parallelogram with four right angles

Rectangle

An isometry that “flips” a figure over a line while reversing orientation

Reflection over a line

A parallelogram with four sides that have the same length

Rhombus

An angle whose degree measure is 90°

Right angle

A triangle that contains a right angle

Right triangle

An isometry that “turns” a figure a specified number of degrees in a given direction (clockwise or counterclockwise) about some fixed point called the center of rotation

Rotation

A figure has rotational symmetry if it coincides with its image for some rotation of 180° or less

Rotational symmetry

Two triangles are congruent if two pairs of corresponding sides are congruent and the angles formed by these sides are congruent

SAS Theorem of Congruence

A triangle in which no two sides have the same length

Scalene triangle

A line that intersects the circle in two different points

Secant of a circle

An arc whose end points are a diameter of the circle

Semicircle

Figures that have the same shape but may have different sizes. Two polygons with the same number of sides are similar if corresponding angles are congruent and the lengths of corresponding sides are in proportion

Similar polygons

The ratio of the length of the leg that is opposite the acute angle to the length of the hypotenuse

Sine of an acute angle of a right triangle

A numerical measure of the steepness of a non-vertical line. The slope of a line is the difference of the coordinates of any two different points on the line divided by the difference of the corresponding x-coordinates of the two points. The slope of a horizontal line is 0, and the slope of a vertical line is undefined

Slope

A formula used to calculate the slope of a non-vertical line when the coordinates of two points on the line are given. The slope, m, of a non-vertical line that contains points x and y and X and Y is given by the formula m = Y – y over X - x

Slope formula

An equation that has the form y = m * x + b, where m is the slope of the line, b is the y-coordinate of the point at which the line crosses the y-axis

Slope-intercept form of an equation of a non-vertical line

A rectangle all of whose sides have the same length

Square

Two triangles are congruent if three pairs of corresponding sides are congruent

SSS Theorem of Congruence

Two angles whose measures add up to 180°

Supplementary angles

Circles in the same plane that are tangent to the same line at the same point

Tangent circles

The ratio of the length of the leg that is opposite a given acute angle to the length of the leg that is adjacent to the same angle

Tangent ratio of an acute angle of a right triangle

A line that intersects the circle in exactly one point, called the point of tangency

Tangent to a circle

A generalization that can be proved

Theorem

A mapping of the elements of two sets where the elements are points such that each point of the object is mapped onto exactly one point called its image and each image point corresponds to exactly one point of the original object called the preimage

Transformation

An isometry that “slides” all points of a figure the same distance in the same direction

Translation

A line that intersects two lines at different points

Transversal

A quadrilateral with exactly one pair of parallel lines

Trapezoid

A polygon with three sides

Triangle

A term that can be described but is so basic that it cannot be defined. The terms point, line, and plane are undefined in geometry

Undefined term

The point at which two sides of the polygon intersect

Vertex (plural: Vertices) of a polygon

The angle formed by the congruent sides of the isosceles triangle

Vertex angle of an isosceles triangle

Pairs of non-adjacent ( opposite ) angles formed by two intersecting lines

Vertical angles

The capacity of a solid figure as measured by the number of cubic units it contains

Volume of a solid figure

The horizontal number line in the coordinate plane

X-axis

The first number in the ordered pair that represents the coordinates of a point in the coordinate plane. The x-coordinate gives the directed horizontal distance of the point from the origin

X-coordinate

The vertical number line in the coordinate plane

Y-axis

The second number in the ordered pair that represents the coordinates of a point in the coordinate plane. The y-coordinate gives the directed vertical distance of the point from the origin

Y-coordinate

The y-coordinate of the point at which a non-vertical line crosses the y-axis

Y-intercept