Geometry – Glossary | Open Up HS Math (TN), Family

A–F

AA similarity theorem

Two triangles are similar if they have two corresponding angles that are congruent.

acute angle

An angle whose measure is between $0 °$ and $90 °$ .

$∠ A B C$ is an acute angle.

acute triangle

A triangle with three acute angles.

Angles $A$ , $B$ , and $C$ are all acute angles.

Triangle $A B C$ is an acute triangle.

adjacent

adjacent angles

Two non-overlapping angles with a common vertex and one common side.

$∠ 1$ and $∠ 2$ are adjacent angles:

alternate exterior angles

A pair of angles formed by a transversal intersecting two lines. The angles lie outside of the two lines and are on opposite sides of the transversal.

See angles made by a transversal.

alternate interior angles

A pair of angles formed by a transversal intersecting two lines. The angles lie between the two lines and are on opposite sides of the transversal.

See also angles made by a transversal.

altitude

Altitude of a triangle:

A perpendicular segment from a vertex to the line containing the base.

Altitude of a solid:

A perpendicular segment from a vertex to the plane containing the base.

Ambiguous Case of the Law of Sines

The Ambiguous Case of the Law of Sines occurs when we are given SSA information about the triangle. Because SSA does not guarantee triangle congruence, there are two possible triangles.

To avoid missing a possible solution for an oblique triangle under these conditions, use the Law of Cosines first to solve for the missing side. Using the quadratic formula to solve for the missing side will make both solutions become apparent.

angle

Two rays that share a common endpoint called the vertex of the angle.

angle bisector

A ray that has its endpoint at the vertex of the angle and divides the angle into two congruent angles.

angle of depression/angle of elevation

Angle of depression: the angle formed by a horizontal line and the line of sight of a viewer looking down. Sometimes called the angle of decline.

Angle of elevation: the angle formed by a horizontal line and the line of sight of a viewer looking up. Sometimes called the angle of incline.

angle of rotation

The fixed point a figure is rotated about is called the center of rotation. If one connects a point in the pre-image, the center of rotation, and the corresponding point in the image, they can see the angle of rotation. A counterclockwise rotation is a rotation in a positive direction. Clockwise is a negative rotation.

angles associated with circles: central angle, inscribed angle, circumscribed angle

Central angle: An angle whose vertex is at the center of a circle and whose sides pass through a pair of points on the circle.

Inscribed angle: An angle formed when two secant lines, or a secant and tangent line, intersect at a point on a circle.

Circumscribed angle: The angle made by two intersecting tangent lines to a circle.

angles made by a transversal

arc length

The distance along the arc of a circle. Part of the circumference.

Equation for finding arc length:

$s = r θ$

Where $r$ is the radius and $θ$ is the central angle in radians.

arc of a circle, intercepted arc

Arc: A portion of a circle.

Intercepted arc: The portion of a circle that lies between two lines, rays, or line segments that intersect the circle.

area model

A rectangular model used to represent the probability of an event. The whole figure represents all possible outcomes of the experiment. The shaded portion of the figure represents the desired outcomes of the event. Area models are particularly useful for representing the probability of experiments that consist of two compound events where the outcomes of either or both events are not equally likely.

For example, if $\frac{1}{4}$ of the population can bend their thumb back, and $\frac{7}{10}$ of the population can roll their tongues, then the probability that a person can bend their thumb and roll their tongue can be represented by a region that is $\frac{1}{4}$ of the length of one side of the rectangle and $\frac{7}{10}$ of the other side.

asymptote

A line that a graph approaches, but does not reach. A graph will never touch a vertical asymptote, but it might cross a horizontal or an oblique (also called slant) asymptote.

Horizontal and oblique asymptotes indicate the general behavior of the ends of a graph in both positive and negative directions. If a rational function has a horizontal asymptote, it will not have an oblique asymptote.

Oblique asymptotes only occur when the numerator of $f (x)$ has a degree that is one higher than the degree of the denominator.

auxiliary line

An extra line or line segment drawn in a figure to help with a proof.

$\overset{\leftrightarrow}{C D}$ is an auxiliary line (added to the diagram of $△ A B C)$ to help prove that the sum of the angles $(m ∠ 4 + m ∠ 2 + m ∠ 5) = 180 °$ .

bisect (verb); bisector (noun) (midpoint)

To divide into two congruent parts.

A bisector can be a point or a line segment.

A perpendicular bisector divides a line segment into two congruent parts and is perpendicular to the segment.

Cavalieri's principle

If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. Therefore, volume formulas for prisms and cylinders work for both right and oblique cylinders and prisms.

center of dilation

See dilation.

central angle

An angle whose vertex is at the center of a circle and whose sides pass through a pair of points on the circle.

centroid

The point of concurrency of a triangle’s three medians.

chord of a circle

A chord of a circle is a straight line segment whose endpoints both lie on the circle. In general, a chord is a line segment joining two points on any curve.

A diameter is a special chord that passes through the center of the circle.

circle

All points in a plane that are equidistant from a fixed point called the center of the circle. The circle is named after its center point. The distance from the center to the circle is the radius. A line segment from the center point to a point on the circle is also called a radius (plural radii, when referring to more than one).

Notation: $⨀ D$

circle: equation in standard form; equation in general form

The standard form of a circle’s equation is ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ where $(h, k)$ , is the center and $r$ is the radius.

The general form of the equation of a circle has ${(x - h)}^{2}$ and ${(y - k)}^{2}$ and $r^{2}$ multiplied out and then like terms have been collected.

$x^{2} + y^{2} + D x + E y + F = 0$

circumcenter

The point where the perpendicular bisectors of the sides of a triangle intersect. The circumcenter is also the center of the triangle’s circumcircle—the circle that passes through all three of the triangle’s vertices.

circumscribe

To draw a circle that passes through all of the vertices of a polygon. The circle is called the circumcircle.

All of these polygons are inscribed in the circles.

clockwise / counterclockwise

clockwise: Moving in the same direction, as the hands on a clock move.

counterclockwise: Moving in the opposite direction, as the hands on a clock move.

coincides (superimposed or carried onto)

When working with transformations, we use words like coincide, superimposed, or carried onto to refer to two points or line segments that occupy the same position on the plane.

collinear, collinearity

When three or more points lie in a line.

Note: Any two points can define a line.

Noncollinear: Not collinear.

complement (in probability)

The complement of an event is the subset of outcomes in the sample space that are not in the event. This means that in any given experiment, either the event or its complement will happen, but not both. The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1.

complementary angles

Two angles whose measures add up to $90 °$ .

compound events

Two or more independent events.

concave and convex

Polygons are either convex or concave.

Convex polygon— no internal angle that measures more than $180 °$ . If any two points are connected with a line segment in the convex polygon, the segment will lie on or inside the polygon.

Concave polygon—at least one internal angle measures more than $180 °$ . If it’s possible to find two points on the polygon that when connected by a line segment, the segment exits the concave polygon.

concentric circles

Circles with a common center.

concurrent lines

A set of two or more lines in a plane are said to be concurrent if they all intersect at the same point. Lines $\overset{\leftrightarrow}{A B}$ , $\overset{\leftrightarrow}{C D}$ , and $\overset{\leftrightarrow}{E F}$ are concurrent lines. They intersect at point $M$ .

Point $M$ is the point of concurrency.

conditional probability

The measure of an event, given that another event has occurred.

The conditional probability of an event $B$ is the probability that the event will occur, given the knowledge that an event $A$ has already occurred. This probability is written $P (B | A)$ , notation for the probability of $B$ given $A$ .

The likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome.

Notation: $P (B | A)$ The probability that event $B$ will occur given the knowledge that event $A$ has already occurred.

In the case where $A$ and $B$ are independent (where event $A$ has no effect on the probability of event $B$ ); the conditional probability of event $B$ given event $A$ is simply the probability of event $B$ , that is, $P (B) .$

If events $A$ and $B$ are not independent, then the probability of the intersection of $A$ and $B$ (the probability that both events occur) is defined by

$P (A and B) = P (A) P (B | A)$

From this definition, the conditional probability $(B | A)$ is obtained by dividing by $P (A)$ :

$P (B | A) = \frac{P (A and B)}{P (A)}, given that P (A) > 0$

conditional statement

A conditional statement (also called an “if-then” statement) is a statement with a hypothesis $P$ , followed by a conclusion $Q$ . Another way to define a conditional statement is to say, “If this happens, then that will happen.” $If P, then Q$ .

The converse of a conditional statement switches the conclusion $Q$ , and the hypothesis $P$ to say: $If Q, then P$ .

A true conditional statement does not guarantee that the converse is true.

Examples: conditional statement: If it rains, the roads will be wet.

Converse: If the roads are wet, then it must have rained.

The converse is not necessarily true. Perhaps a pipe broke and flooded the road.

cone: right, oblique

A 3-D figure that has length, width, and height. A cone has a single flat face (also called its base) that’s in the shape of a circle. The body of the cone has curved sides that lead up to a narrow point at the top called a vertex or an apex.

A right cone has a vertex that is directly over the center of the base. In an oblique cone the vertex is not over the center of the base.

congruence statement

A mathematical statement that uses the $≅$ symbol. Examples:

$\begin{aligned} △ A B C ≅ △ X Y Z & Triangle A B C is congruent to triangle X Y Z . \\ \overset{―}{A B} ≅ \overset{―}{E F} & Line segment A B is congruent to line segment E F . \\ ∠ R ≅ ∠ T & Angle R is congruent to angle T . \end{aligned}$

Only figures or shapes can be congruent. Numbers are equal.

congruent (CPCTC)

Two triangles (figures) are congruent if they are the same size and same shape. Two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other.

The symbol for congruent is $≅$ .

If it’s given that two triangles (figures) are congruent, then the Corresponding Parts of the Congruent Triangles (figures) are Congruent (CPCTC).

conjecture

A mathematical statement that has not yet been rigorously proven. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. When a conjecture is proven, it becomes a theorem.

construction

Creating a diagram of geometric figures and items such as perpendicular lines or a regular pentagon using only a compass and straightedge.

A construction yields an exactly reproducible and unambiguous result, of which all properties can be measured as expected (within the accuracy of the instruments use.)

Constructing an angle bisector:

convergence

Moving toward or approaching a definite value or point.

converse statement

See conditional statement.

corresponding angles

Angles that are in the same relative position.

corresponding parts (in a triangle)

The word corresponding refers to parts that match between two congruent figures. Corresponding angles and corresponding sides will have the same measurements in congruent figures.

corresponding points / sides

Points, sides, and angles can all be corresponding. It means they are in the same relative position.

counterexample

An example that disproves a statement or conjecture. One counterexample can disprove a conjecture based on many examples.

Statement: All blondes drive red cars.

Counterexample: My mom is blonde, but her car is silver.

CPCTC

See congruent (CPCTC).

cross-section of a solid

The face formed when a three-dimensional object is sliced by a plane.

cyclic polygon

A polygon that can be inscribed in a circle. All of the vertices of the polygon lie on the same circle.

cylinder: right, oblique

In a right cylinder, the sides make a right angle with the two bases.

In an oblique cylinder, the bases remain parallel to each other, but the sides lean over at an angle that is not $90 °$ .

definition

A statement of the meaning of a word or symbol that is accepted by the mathematical community. A good mathematical definition uses previously defined terms and the symbol that represents it. Once a word has been defined, it can be used in subsequent definitions.

degree

A degree is the measure of an angle of rotation that is equal to $\frac{1}{360}$ of a complete rotation around a fixed point. A measure of $60$ degrees would be written as $60 °$ .

density

In science, density describes how much space an object or substance takes up (its volume) in relation to the amount of matter in that object or substance (its mass). If an object is heavy and compact, it has a high density. If an object is light and takes up a lot of space, it has a low density.

$density = \frac{mass}{volume}$

Density can also refer to how many people are crowded into a small area or how many trees are growing in a small space or a large space. In that sense it is a comparison of compactness to space.

$density = \frac{how many}{space or area}$

diagonal

Any line segment that connects nonconsecutive vertices of a polygon.

dilation

A transformation that produces an image that is the same shape as the pre-image but is of a different size. A description of a dilation includes the scale factor and the center of dilation.

A dilation is a transformation of the plane, such that if $O$ is the center of the dilation and a nonzero number $k$ is the scale factor, then $P^{'}$ is the image of point $P$ , if $O$ , $P$ , and $P^{'}$ are collinear and $\frac{O P^{^{'}}}{O P} = k$ .

directed distance

Distance is always positive. A directed distance has length and direction. Partitions occur on line segments that are referred to as directed line segments. A directed segment is a segment that has distance (length) and direction. It is important to understand that a directed segment has a starting point referred to as the initial point and a direction from which to move away from the starting point. This will clarify the location of the partition ratio on the segment.

direction of a vector

The direction of a vector is determined by the angle it makes with a horizontal line.

See vector.

directrix

See parabola.

disc or disk

See solid of revolution.

disjoint

See mutually exclusive.

edge / face / vertex of a 3-D solid

Edge: The line that is the intersection of two planes.

Face: A flat surface on a $3$ -D solid.

Vertex: (pl. vertices) Each point where two or more edges meet; a corner.

ellipse

An ellipse is the set of all points $(x, y)$ in a plane that have the same total distance from two fixed points called the foci.

The distance from the point on the ellipse to each of the two foci is labeled $d 1$ and $d 2$ .

Equation of an ellipse with center $(h, k)$ , $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

equality statements

A mathematical sentence that states two values are equal.

It contains an equal sign.

equidistant

A shortened way of saying equally distant; the same distance from each other or in relation to other things.

equilateral, equilateral triangle

Equilateral means equal side lengths.

In an equilateral triangle, all of the sides have the same length.

event

A set of outcomes of an experiment. An event is a subset of all possible outcomes.

For example, if we roll a 6-sided number cube there are six possible outcomes. The event of rolling an even number would consist of the outcomes of rolling a 2, 4 or 6. Rolling the dice is the experiment. Getting an even number is the event.

There are three kinds of events. Events can be:

Independent (each event is not affected by other events),
Dependent (also called “Conditional”, where an event is affected by other events)
Mutually Exclusive (events can’t occur at the same time).

exterior angle of a triangle (remote interior angles)

An exterior angle of a triangle is an angle formed by one side of the triangle and the extension of an adjacent side of the triangle. There are two exterior angles at every vertex of a triangle.

exterior angle theorem

The measure of an exterior angle in any triangle is equal to the sum of the two remote interior angles.

false negative/positive

The result of a test that appears negative when it should not. An example of a false negative would be if a particular test designed to detect cancer returns a negative result, but the person actually does have cancer.

A false positive is where you receive a positive result for a test, when you should have received a negative result.

flow proof

See proof: types—flow, two-column, paragraph.

focus

See parabola.

frustum

The part of a solid such as a pyramid or a cone that remains after cutting off a top portion with a plane parallel to the base.

G–L

geometric mean

A special type of average where $n$ numbers are multiplied together and then the $n th$ root is taken. For two numbers, the geometric mean would be the square root. For three numbers, it would be the cube root.

Example: The geometric mean of $4$ and $9$ is $\sqrt{4 \cdot 9} = 6$ .

The geometric mean of two numbers $a$ and $c$ is the number $b$ such that $\frac{a}{b} = \frac{b}{c}$ .

geometric series

The sum of the terms in a geometric sequence represented by summation notation $\sum_{k = 1}^{n} f (k)$ .

Example: $\sum_{k = 1}^{4} 3 k = (3 \cdot 1) + (3 \cdot 2) + (3 \cdot 3) + (3 \cdot 4) = 30$

hexagon

A six-sided polygon.

hyperbola

A hyperbola is the set of all points such that the difference of the distances to the foci is constant.

Equation: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

hypotenuse

The longest side in a right triangle.

The side opposite the right angle.

image

A picture; a visual representation of a thing. See pre-image / image.

incenter

The point of intersection of the angle bisectors in a triangle is the incenter. Each point on the angle bisector is equidistant from the sides of the angle.

The point at which all the three angle bisectors meet is the center of the incircle.

independent event / dependent event

When two events are said to be independent of each other, the probability that one event occurs in no way affects the probability of the other event occurring.

When you flip two coins, each flip is an independent event.

An event is dependent if the occurrence of the first event affects the occurrence of the second so that the probability is changed.

Example: Suppose there are $5$ balls in a box. What is the chance of getting a green ball out of the box on the first try? A green ball is selected and removed in event $1$ . What is the chance of getting a green ball on the second try?

inscribed angle

See angles associated with circles.

inscribed in a circle

intersection of sets

The intersection of two sets $X$ and $Y$ , is the set containing all of the elements of $X$ that also belong to $Y$ . The symbol for intersection is $\cap$ .

For example: If $X = {5, 6, 7, 8, 9, 10}$ and $Y = {5, 10, 15}$ then $X \cap Y = {5, 10}$ .

inverse trigonometric ratio

The inverse of a trigonometric function is used to obtain the measure of an angle when the trigonometric ratio is known.

Example: The inverse of sine is denoted as arcsine, or on a calculator it will appear as $\sin^{- 1}$ .

If $\sin A = \frac{\sqrt{3}}{2}$ and the measure of the angle $A$ is needed, write $\sin^{- 1} (\frac{\sqrt{3}}{2})$ to express this. The answer to the expression is the measure of the angle.

$\sin^{- 1} (\frac{\sqrt{3}}{2}) = 60 °$

All of the inverse trigonometric functions are written the same way.

$\sin^{- 1} (\frac{o}{h}) = A, \cos^{- 1} (\frac{a}{h}) = A, \tan^{- 1} (\frac{o}{a}) = A$

isosceles triangle, trapezoid

The word isosceles is only used to describe a triangle or a trapezoid with two congruent sides.

joint events

Events that can occur at the same time.

Two-way tables show joints. See two-way tables.

kite

A quadrilateral with two pairs of congruent, adjacent sides.

law of cosines

For any triangle with angles $A$ , $B$ , and $C$ , and sides of lengths $a$ , $b$ , and $c$ , where $a$ is opposite $∠ A$ , and $b$ is opposite $∠ B$ and $c$ is opposite $∠ C$ , these equalities hold true:

$a^{2} = b^{2} + c^{2} - 2 a b \cos A$

$b^{2} = a^{2} + c^{2} - 2 a c \cos B$

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$

The law of cosines is useful for finding:

the third side of a triangle when we know two sides and the angle between them.
the angles of a triangle when we know all three sides.

law of sines

limit (convergence)

Sometimes in math we can see that an output is getting closer and closer to a value. We can also see that the output won’t exceed this value. We call this a limit.

Example 1: As $n$ gets larger, the value of $n \cdot \sin (\frac{360 °}{2 n}) \cdot \cos (\frac{360 °}{2 n})$ is getting very close to the value of $π$ . We say $π$ is the limit.

Example 2: The more sides in a polygon, the closer the polygon gets to being a circle. The circle is the limit.

More formally: A repeated calculation process that approaches a unique value, called the limit.

line

A line is an undefined term because it is an abstract idea, rather than concrete like a stroke of ink. It is defined as a line of points that extends infinitely in two directions. It has one dimension, length. Points that are on the same line are called collinear points. A line is defined by two points, such as line $A B$ .

Notation: $\overset{\leftrightarrow}{A B}$

line of symmetry

The vertical line that divides the graph into two congruent halves, sometimes called axis of symmetry.

The equation for the line of symmetry in a coordinate plane is always:

$x = the number that the line of symmetry passes through$

line segment

A piece of a line with two endpoints.

Notation: $\overset{―}{A B}$ represents the line segment with endpoints at point $A$ and point $B$ . $\overset{―}{A B}$ is an object.

A line segment has length and can be measured.

The notation $A B$ (without any kind of line above it) refers to the length of segment $A B$ .

linear pair

Two supplementary angles that share a vertex and a side.

A linear pair always make a line.

M–R

magnitude of a vector

The length of a vector.

See vector.

major axis, minor axis of an ellipse

The major axis is the longest diameter of an ellipse. It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse. $C D$ is the major axis.

The minor axis is the shortest diameter (at the narrowest part of the ellipse).

median in a triangle

A line segment in a triangle that extends from any vertex to the midpoint of the opposite side.

midline of a triangle

$\overset{―}{D E}$ is the midline of $△ A B C$ .

midline of a triangle theorem

The midline of a triangle or the midsegment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.

midpoint

A point on a line segment that divides it into two equal parts.

The formula for finding half the distance between two points (or the midpoint $M$ ) in a coordinate grid is:

$M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) .$

$M = (\frac{- 3 + 2}{2}, \frac{- 2 + 4}{2}) = (- 0.5, 1)$

See also bisect.

model, mathematical

Modeling with mathematics is the practice of making sense of the world through a mathematical perspective. A mathematical model could be an equation, graph, diagram, formula, sketch, computer program, or other representation that will help you to study different components of a function or to make predictions about behavior.

mutually exclusive

Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint. If two events are disjoint, then the probability of them both occurring at the same time is 0.

mutually exclusive event

Both events can’t happen at the same time. It must be one or the other, but not both.

Example: heads and tails are mutually exclusive when flipping a coin.

n-gon

A polygon with $n$ number of sides.

See polygon.

obtuse angle / obtuse triangle

octagon

An eight-sided polygon.

opposite (or negative) reciprocal slope

Slopes of perpendicular lines are opposite reciprocals, so that the product of the slopes is $- 1$ . (See perpendicular lines)

opposite angles, opposite vertices

Opposite angles in a quadrilateral do not share a side.

A vertex (plural, vertices) is part of an angle.

opposite side in a triangle

A side opposite an angle in a triangle is the side that is not part of the angle.

opposite sides (in a parallelogram or an even-sided polygon)

If two sides in a parallelogram are parallel, they must be opposite sides.

If two sides in an even-sided polygon are parallel, they must be opposite sides.

orientation

The orientation is determined by the order in which a figure’s vertices are labeled. In the diagram, the vertices of the green pentagon are labeled from $J$ to $K$ to $L$ to $M$ to $N$ in a clockwise direction.

In the blue pentagon, the orientation of the vertices has changed. The corresponding vertices go in a counterclockwise direction from $J^{'}$ to $K^{'}$ to $L^{'}$ to $M^{'}$ to $N^{'}$ .

origin

The origin is a starting point. The coordinates for every other point are based on how far that point is from the origin. At the origin, both $x$ and $y$ are equal to zero, and the $x$ -axis and the $y$ -axis intersect.

parabola: conic definition, geometric definition

A parabola is the set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix . The directrix is perpendicular to the axis of symmetry of a parabola and does not touch the parabola.

parallel line

parallelogram

A quadrilateral in which the opposite sides are parallel.

pentagon

A five-sided polygon.

perpendicular bisector

The line (line segment or ray) that divides a line segment into two equal lengths and makes a right angle with the line segment it divides.

perpendicular lines

Two lines or line segments are perpendicular if they have opposite, reciprocal slopes, or if one is vertical and the other is horizontal. Two lines are perpendicular if their intersection forms four right ( $90 °$ ) angles.

plane

A plane is an undefined term because it is an abstract idea rather than concrete like a piece of paper. A plane has two dimensions. It can be identified by determining three noncollinear points. It is labeled according to the letters used to label the points, such as plane $A B C$ .

point

A point is an undefined term because it is an abstract idea rather than concrete like a dot. A point in geometry is a location. It has no size, (i.e., no width, no length, and no depth). A point is labeled with a dot and a capital letter.

point of concurrency

See concurrent lines.

polygon

Any 2-D shape formed with line segments that connect at their endpoints, making a closed figure. The location where any two line segments connect is called a vertex.

Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. The name identifies how many sides the shape has. For example, a triangle has three sides, a quadrilateral has four sides, a pentagon five sides, and an octagon eight sides. A regular polygon is made up of congruent line segments.

In a regular polygon, all sides are congruent, and all angles are congruent.

postulate

A simple and useful statement in geometry that is accepted by the mathematical community as true without proof.

pre-image / image

The pre-image is the original figure. The image is the new figure created from the pre-image through a sequence of transformations or a dilation.

preserves distance and angle measure

Measurements are not changed under a rigid transformation.

prism: right, oblique

Prism: Also called a polyhedron.

A solid object with two identical ends and flat sides. The ends (bases) are parallel. The shape of the ends gives the prism its name, such as triangular prism or square prism. The sides are parallelograms.

proof by contradiction

A way to justify a claim is to use a proof by contradiction method, in which one assumes the opposite of the claim is true, and shows that this leads to a contradiction of something that is known to be true.

proof: types—flow, two-column, paragraph

properties of equality

The properties of equality describe operations that can be performed on each side of the equal sign ( $=$ ) and still ensure that the expressions remain equivalent.

In the table below, $a$ , $b$ , and $c$ stand for arbitrary numbers in the rational, real, or complex number systems. The properties of equality are true in these number systems.

Reflexive property of equality	$a = a$
Symmetric property of equality	If $a = b$ , then $b = a$
Transitive property of equality	If $a = b$ and $b = c$ , then $a = c$
Addition property of equality	If $a = b$ , then $a + c = b + c$
Subtraction property of equality	If $a = b$ , then $a - c = b - c$
Multiplication property of equality	If $a = b$ , then $a \times c = b \times c$
Division property of equality	If $a = b$ and $c \neq 0$ , then $a \div c = b \div c$
Substitution property of equality	If $a = b$ , then $b$ may be substituted for $a$ in any expression containing $b$

proportion: proportionality statement

A proportion is a statement that two ratios are equal.

pyramid

A 3-D shape that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex.

Pythagorean theorem

The relationship among the lengths of the sides of a right triangle that results in the sum of the squares of the lengths of the legs equaling the square of the length of the hypotenuse.

quadrilaterals: types

A quadrilateral is a four-sided polygon. See the diagram for various types of quadrilaterals.

quantity

A quantity is an amount, number, or measurement. It answers the question “How much?”

radian

A unit of measure for angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

The ratio of the length of an intercepted arc to the radius of the circle on which that arc lies.

radii

Plural of radius. See circle.

ratio

A ratio compares the size or amount of two values.

Here is a sentence that compares apples to oranges as shown in the diagram below: “We have five apples for every three oranges.” It describes a ratio of $5$ to $3$ or $5 : 3$ . A ratio can also be written as a fraction, in this case $\frac{5}{3}$ .

Compare oranges to apples. The ratio changes to $3 : 5$ or $\frac{3}{5}$ .

The two previous ratios are called part-to-part ratios. Another way to write a ratio is to compare a part to a whole.

Compare apples to the total amount of fruit. The ratio changes to $5 : 8$ or $\frac{5}{8}$ .

Ratios can be scaled up or down. There are $7$ bags of fruit, each containing $3$ oranges and $5$ apples. The ratio $5$ to $3$ still represents the number of apples compared to the number of oranges. But the ratio $35$ to $21$ also compares the number of apples to the number of oranges.

ray

A part of a line that has a fixed starting point (endpoint), and then continues toward infinity.

Notation: $\vec{A B}$ – ray $A B$

A ray is named using its endpoint first, and then any other point on the ray.

reasoning – deductive/inductive

Two Types of Reasoning
Inductive reasoning: from a number of observations, a general conclusion is drawn.	Deductive reasoning: from a general premise (something we know), specific results are predicted.
Observations	General Premise
Each time I make two lines intersect, the opposite angles are congruent. I have tried this 20 times and it seems to be true. Conclusion: Opposite angles formed by intersecting lines are always congruent.	Given: Angles 1, 2, 3, and 4 are formed by two intersecting lines. Prove: Opposite angles formed by intersecting lines are always congruent.

Two Types of Reasoning

Inductive reasoning:

from a number of observations, a general conclusion is drawn.

Deductive reasoning:

from a general premise (something we know), specific results are predicted.

Observations

General Premise

Each time I make two lines intersect, the opposite angles are congruent. I have tried this 20 times and it seems to be true.

Conclusion:

Opposite angles formed by intersecting lines are always congruent.

Given: Angles 1, 2, 3, and 4 are formed by two intersecting lines.

Prove: Opposite angles formed by intersecting lines are always congruent.

rectangle

See quadrilaterals: types.

reference angle

The acute angle between the terminal ray of an angle in standard position and the $x$ -axis.

reflection

A reflection is a rigid transformation (isometry). In a reflection, the pre-image and image points are the same distance from the line of reflection; the segment connecting corresponding points is perpendicular to the line of reflection.

The orientation of the image is reversed.

regular polygon

See polygon.

rhombus

A quadrilateral in which all sides are congruent.

right angle

An angle that measures $90 °$ .

The symbol for a right angle in a geometric figure is a box.

rigid transformation

Also called an isometry. The word rigid means that the pre-image and image are congruent. The rigid transformations include translation, rotation, and reflection.

rotation

A rotation is a rigid transformation. In a rotation, all points remain the same distance from the center of rotation, move in the same direction, and through the same central angle. The orientation of the pre-image remains the same.

rotational symmetry

See symmetry.

S–X

same-side interior angles

See angles made by a transversal.

SAS triangle similarity

See triangle similarity.

scalar quantity

A scalar quantity is usually depicted by a number, numerical value, or a magnitude, but no direction.

scale factor

The ratio of any two corresponding lengths in two similar geometric figures.

scalene triangle

A triangle that has three unequal sides.

secant line (in a circle), tangent line

Secant line: A line that intersects a circle at exactly two points.

Tangent line: A line that intersects a circle at exactly one point.

sector

The part of a circle enclosed by two radii of a circle and their intercepted arc.

A pie-shaped part of a circle.

segment of a circle

A segment of a circle is a region in a plane that is bounded by an arc of a circle and by the chord connecting the endpoints of the arc.

side-splitter theorem

The side-splitter theorem is related to the midline of a triangle theorem. It extends the rule to say if a line intersects two sides of a triangle and is parallel to the third side of the triangle, it divides those two sides proportionally.

similarity

A 2-D figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

solids of revolution

A 3-D object formed by spinning a 2-D figure about an axis.

A disk is a slice of the solid of revolution. Each disk’s face is a circle.

A washer is a slice of a hollow solid of revolution. Its face is a circle with a hole in the center.

special right triangles

There are two special right triangles. They are special because they can be solved without using trigonometry.

square

See quadrilaterals: types.

SSS triangle similarity

See triangle similarity.

straight angle

When the legs of an angle are pointing in exactly opposite directions, the two legs form a single straight line through the vertex of the angle. The measure of a straight angle is always $180 °$ . It looks like a straight line.

summation notation

supplementary angles

Two angles whose measures add up to exactly $180 °$ .

symmetric

If a figure can be folded or divided in half so that the two halves match exactly, then such a figure is called a symmetric figure. The fold line is the line of symmetry.

symmetry

A line that reflects a figure onto itself is called a line of symmetry.

A figure that can be carried onto itself by a rotation is said to have rotational symmetry.

tessellation

A tessellation is a regular pattern made up of flat shapes repeated and joined together without any gaps or overlaps. Many regular polygons tessellate, meaning they can fit together without any gaps.

theorem

A theorem is a statement that can be demonstrated to be true by using definitions, postulates, properties, and previously proven theorems.

The process of showing a theorem to be correct is called a proof.

theoretical probability

A prediction of the probability of an event based on mathematical models rather than experimentation. When using a Probability Area Model, the probability of an event is given by the ratio of the shaded area representing the event to the total area ofnt occurs by the total number of outcomes in the sample space.

translation

A translation is a rigid transformation.

transversal

A line that passes through two lines in the same plane at two distinct points. The two lines do not need to be parallel. But when the lines are parallel, several special angle relationships are formed.

trapezoid

A quadrilateral with exactly one pair of parallel opposite sides.

(Note: A trapezoid can also be defined as a quadrilateral with at least one pair of opposite sides that are parallel. This definition makes it possible for parallelograms to be a special type of trapezoid.)

In an isosceles trapezoid, the two opposite sides that are not parallel are congruent and form congruent angles with the parallel sides. This feature of an isosceles trapezoid only exists if the trapezoid is not a parallelogram.

tree diagram

A tool in probability and statistics used to calculate the number of possible outcomes of an event, as well as list those possible outcomes in an organized manner.

triangle congruence criteria: ASA, SAS, AAS, SSS

Two triangles are congruent if all three sides and all three angles are congruent. But sometimes only three pieces of information are sufficient to prove two triangles congruent.

ASA stands for “angle-side-angle.”

SAS stands for “side-angle-side.”

AAS stands for “angle-angle-side.”

SSS stands for “side-side-side.”

triangle similarity

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. Similar triangles are the same shape, but not necessarily the same size.

There are three similarity patterns that provide sufficient information to prove two triangles are similar:

AA Similarity

SAS Similarity

SSS Similarity

trigonometric ratios in right triangles: sine A, cosine A, tangent A

An operation that relates the measure of an angle with a ratio of the lengths of the sides in a right triangle. There are three trigonometric ratios, plus their reciprocals. See Reciprocal trigonometric functions for definitions.

$sine A$ abbreviated $\sin A$

$cosine A$ abbreviated $\cos A$

$tangent A$ abbreviated $\tan A$

A trigonometric ratio always includes a reference angle.

In right triangle $A B C$ , the trigonometric ratios are defined as:

$\sin A = \frac{the length of the opposite side}{the length of the hypotenuse} = \frac{a}{c}$

$\cos A = \frac{the length of the adjacent side}{the length of the hypotenuse} = \frac{b}{c}$

$\tan A = \frac{the length of the opposite side}{the length of the adjacent side} = \frac{a}{b}$

Note that each trigonometric function above references the angle $A$ . If angle $B$ was referenced as the angle, the opposite and adjacent sides would be in reference to angle $B$ , and they would switch sides.

two-column proof

See proof:types—flow, two-column, paragraph.

two-way table

A table listing two categorical variables whose values have been paired such that the possible values of one variable make up the rows and the possible values for the other variable make up the columns. The green cells on this table are where the joint frequency numbers are located. They are called joint frequency because you are joining one variable from the row and one variable from the column. The marginal frequency numbers are the numbers on the edges of a table. On this table, the marginal frequency numbers are in the purple cells.

union

The union of two sets is a set containing all elements that are in set $A$ or in set $B$ (or possibly both). The symbol for union is $\cup$ .

For example, ${5, 8, 11} \cup {6, 7, 9, 10} = {5, 6, 7, 8, 9, 10, 11}$ .

vector, vector quantity

A vector is a quantity that has magnitude (length) and direction.

Notation: $\vec{A B}$

Unlike a geometric ray, a vector has a specific length.

The magnitude $u$ of a vector is calculated using the Pythagorean theorem.

$| u | = \sqrt{1^{2} + 3^{2}} = \sqrt{10}$

The direction of a vector is determined by the angle it makes with a horizontal line. In the diagram, the direction will be represented by theta, $θ$ (or $∠ A$ ). The value of the angle will be found using trigonometry. $\tan^{- 1} (\frac{3}{1}) \approx 71.6 °$

$\vec{A B} = \sqrt{10}$ at $71.6 °$ above the horizontal.

vertical angles

The angles opposite each other when two lines cross. They are always congruent.

washer

A washer is a slice of a hollow solid of revolution. Its face is a circle with a hole in the center.

A–F

G–L

M–R

S–X

Y–Z