sasha vasserman. two triangles are similar if two pairs of corresponding angles are congruent
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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
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