Geometry Definitions


acute triangle	close window
An acute triangle is a triangle with all three angles less than 90°.



altitude	close window
An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.



angle-angle-angle (AAA) similarity	close window
The angle-angle-angle (AAA) similarity test says that if two triangles have corresponding angles that are congruent, then the triangles are similar. Because the sum of the angles in a triangle must be 180°, we really only need to know that two pairs of corresponding angles are congruent to know the triangles are similar.



angle bisector	close window
An angle bisector is a ray that cuts the angle exactly in half, making two equal angles.


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central angle	close window
A central angle is an angle with its vertex at the center of a circle.



centroid	close window
The centroid of a triangle is the point where the three medians meet. This point is the center of mass for the triangle. If you cut a triangle out of a piece of paper and put your pencil point at the centroid, you could balance the triangle.



circle	close window
A circle is the set of all points in a plane that are equidistant from a given point in the plane, which is the center of the circle.



circumcenter	close window
The circumcenter of a triangle is the point where the three perpendicular bisectors meet. This point is the same distance from each of the three vertices of the triangles.



concave polygon	close window
A concave polygon is any polygon with an angle measuring more than 180°. Concave polygons look like they are collapsed or have one or more angles dented in.



concurrent	close window
When three or more lines meet at a single point, they are said to be concurrent. In a triangle, the three medians, three perpendicular bisectors, three angle bisectors, and three altitudes are each concurrent.



congruent	close window
Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they "are the same size and shape," though they may have different orientation. (One might be rotated or flipped compared to the other.)



congruent triangles	close window
Congruent triangles are triangles that have the same size and shape. In particular, corresponding angles have the same measure, and corresponding sides have the same length.



converse	close window
Converse means the "if" and "then" parts of a sentence are switched. For example, "If two numbers are both even, then their sum is even" is a true statement. The converse would be "If the sum of two numbers is even, then the numbers are even," which is not a true statement.



convex polygon	close window
A convex polygon is any polygon that is not concave.



coordinates	close window
Points are geometric objects that have only location. To describe their location, we use coordinates. We begin with a standard reference frame (typically the x- and y-axes). The coordinates of a point describe where it is located with respect to this reference frame. They are given in the form (x,y) where the x represents how far the point is from 0 along the x-axis, and the y represents how far it is from 0 along the y-axis. The form (x,y) is a standard convention that allows everyone to mean the same thing when they reference any point.



cosine	close window
If angle A is an acute angle in a right triangle, the cosine of A is the length of the side adjacent to angle A, divided by the length of the hypotenuse of the triangle. We often abbreviate this as cos A = (adjacent)/(hypotenuse).



cross section	close window
A cross section is the face you get when you make one slice through an object.


diameter	close window
A circle's diameter is a segment that passes through the center and has its endpoints on the circle.


edge	close window
An edge is a line segment where two faces intersect.



equilateral triangle	close window
An equilateral triangle is a triangle with three equal sides.


face	close window
A face is a polygon by which a solid object is bound. For example, a cube has six faces. Each face is a square.



frieze pattern	close window
A frieze pattern is an infinite strip containing a symmetric pattern.


glide reflection	close window
A glide reflection is a combination of two transformations: a reflection over a line followed by a translation in the same direction as the line.


hypotenuse	close window
The hypotenuse in a right triangle is the side of the triangle that is opposite to the right angle.


incenter	close window
The incenter of a triangle is the point where the three angle bisectors meet. This point is the same distance from each of the three sides of the triangle.



inscribed angle	close window
An inscribed angle is an angle whose vertex is on a circle and whose rays intersect the circle.



intercept	close window
An intercept is an intersection of a graph with one of the axes. An intersection with the horizontal axis is often referred to as an x-intercept, and an intersection with the vertical axis is often referred to as a y-intercept.



irregular polygon	close window
An irregular polygon is any polygon that is not regular.



isosceles trapezoid	close window
An isosceles trapezoid is a quadrilateral with one pair of parallel sides and congruent base angles, or it is a trapezoid with congruent base angles.



isosceles triangle	close window
An isosceles triangle is a triangle with two equal sides.


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kite	close window
A kite is a quadrilateral that has two pairs of adjacent sides congruent (the same length).


line	close window
A line has only one dimension: length. It continues forever in two directions (so it has infinite length), but it has no width at all. A line connects two points via the shortest path, and then continues on in both directions.



line segment	close window
A line segment is the portion of a line lying strictly between two points. It has a finite length and no width.



line symmetry or reflection symmetry	close window
A polygon has line symmetry, or reflection symmetry, if you can fold it in half along a line so that the two halves match exactly. The folding line is called the line of symmetry.


median	close window
A median is a segment connecting any vertex of a triangle to the midpoint of the opposite side.



midline	close window
A midline is a segment connecting two consecutive midpoints of a triangle.



midline theorem	close window
The midline theorem states that a midline of a triangle creates a segment that is parallel to the base and half as long.


net	close window
A net is a two-dimensional representation of a three-dimensional object.


obtuse triangle	close window
An obtuse triangle is a triangle with one angle more than 90°.



orthocenter	close window
The orthocenter of a triangle is the point where the three altitudes meet, making them concurrent.

parallel lines

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Parallel lines are two lines in the same plane that never intersect. Another way to think about parallel lines is that they are "everywhere equidistant." No matter where you measure, the perpendicular distance between two parallel lines is constant.

parallelogram

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A parallelogram is a quadrilateral that has two pairs of opposite sides that are parallel.

perpendicular bisector

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The perpendicular bisector of a line segment is perpendicular to that segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.

plane

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A plane is a flat, two-dimensional object. We often represent a plane by a piece of paper, a blackboard, or the top of a desk. In fact, none of these is actually a plane, because a plane must continue infinitely in all directions and have no thickness at all. A plane can be defined by two intersecting lines or by three non-collinear points.

Platonic solid

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A Platonic solid is a solid such that all of its faces are congruent regular polygons and the same number of regular polygons meet at each vertex.

point

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A point specifies only location; it has no length, width, or depth. We usually represent a point with a dot on paper, but the dot we make has some dimension, while a true point has dimension 0.

polygon

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A polygon is a two-dimensional geometric figure with these characteristics:

•	It is made of straight line segments.
•	Each segment touches exactly two other segments, one at each of its endpoints.
•	It is closed -- it divides the plane into two distinct regions, one inside and the other outside the polygon.

polyhedron

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A polyhedron is a closed three-dimensional figure. All of the faces are made up of polygons.

Pythagorean theorem

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The Pythagorean theorem states that if you have a right triangle, then the square built on the hypotenuse is equal to the sum of the squares built on the other two sides. a² + b² = c².


quadrilateral	close window
A quadrilateral is a polygon with exactly four sides.


radius	close window
The radius of a circle is the distance from the circle's center to a point on the circle, and is constant for a given circle.



ray	close window
A ray can be thought of as a half a line. It has a point on one end, and it extends infinitely in the other direction.



rectangle	close window
A rectangle is a quadrilateral with four right angles.



reflection	close window
Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line that acts like the mirror. In reflection, the object changes its orientation (top and bottom, left and right). Depending on the location of the mirror line, the object may also change location.



regular polygon	close window
A regular polygon has sides that are all the same length and angles that are all the same size.



rhombus	close window
A rhombus is a quadrilateral that has all four sides congruent.



right triangle	close window
A right triangle is a triangle with one right (90°) angle.



rotation	close window
Rotation is a rigid motion, meaning an object changes its position but not its size or shape. In a rotation, an object is turned about a "center" point, through a particular angle. (Note that the "center" of rotation is not necessarily the "center" of the object or even a point on the object.) In a rotation, the object changes its orientation (top and bottom). Depending on the location of the center of rotation, the object may also change location.



rotation symmetry	close window
A figure has rotation symmetry if you can rotate (or turn) that figure around a center point by fewer than 360° and the figure appears unchanged.


scalene triangle	close window
A scalene triangle is a triangle with all three sides unequal.



side-angle-side (SAS) congruence	close window
Side-angle-side (SAS) congruence states that if any two sides of a triangle are equal in length to two sides of another triangle and the angles bewteen each pair of sides have the same measure, then the two triangles are congruent; that is, they have exactly the same shape and size.



side-angle-side (SAS) similarity	close window
The side-angle-side (SAS) similarity test says that if two triangles have two pairs of sides that are proportional and the included angles are congruent, then the triangles are similar.



side-side-side (SSS) congruence	close window
The side-side-side (SSS) congruence states that if the three sides of one triangle have the same lengths as the three sides of another triangle, then the two triangles are congruent.



side-side-side (SSS) similarity	close window
The side-side-side (SSS) similarity test says that if two triangles have all three pairs of sides in proportion, the triangles must be similar.



similar	close window
Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.



similar triangles	close window
Similar triangles are triangles that have the same shape but possibly different size. In particular, corresponding angles are congruent, and corresponding sides are in proportion.



sine	close window
If angle A is an acute angle in a right triangle, the sine of A is the length of the side opposite to angle A divided by the length of the hypotenuse of the triangle. We often abbreviate this as sin A = (opposite)/(hypotenuse).



square	close window
A square is a regular quadrilateral.



symmetry	close window
A design has symmetry if you can move the entire design by either rotation, reflection, or translation, and the design appears unchanged.


tangent	close window
If angle A is an acute angle in a right triangle, the tangent of A is the length of the side opposite to angle A divided by the length of the side adjacent to angle A. We often abbreviate this as tan A = (opposite)/(adjacent).



tangram	close window
A tangram is a seven-piece puzzle made from a square. A typical tangram set contains two large isosceles right triangles, one medium isosceles right triangle, two small isosceles right triangles, a square, and a parallelogram.



theorem	close window
A theorem in mathematics is a proven fact. A theorem about right triangles must be true for every right triangle; there can be no exceptions. Just showing that an idea works in several cases is not enough to make an idea into a theorem.



translation	close window
Translation is a rigid motion, meaning an object changes its position but not its size or shape. In a translation, an object is moved in a given direction for a particular distance. A translation is therefore usually described by a vector, pointing in the direction of movement and with the appropriate length. In translation, the object changes its location, but not its orientation (top and bottom, left and right).


translation symmetry	close window
Translation symmetry can be found only on an infinite strip. For translation symmetry, you can slide the whole strip some distance, and the pattern will land back on itself.


transversal	close window
A transversal is a line that passes through (transverses) two other lines. We often consider what happens when the two other lines are parallel to each other.


trapezoid	close window
A trapezoid is a quadrilateral that has one pair of opposite sides that are parallel.


triangle inequality	close window
The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.


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van Hiele levels	close window
Van Hiele levels make up a theory of five levels of geometric thought developed by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof. The levels are (0) visualization, (1) analysis, (2) informal deduction, (3) deduction, and (4) rigor. The theory is useful for thinking about what activities are appropriate for students, what activities prepare them to move to the next level, and how to design activities for students who may be at different levels.



vector	close window
A vector can be used to describe a translation. It is drawn as an arrow. The arrowhead points in the direction of the translation, and the length of the vector tells you the length of the translation.



Venn diagram	close window
A Venn diagram uses circles to represent relationships among sets of objects



vertex	close window
A vertex is the point where two sides of a polygon meet.


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