Math 2 – Glossary | Open Up HS Math (NC), Family

A–F

AA similarity theorem

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

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.

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 made by a transversal

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 $(m ∠ 4 + m ∠ 2 + m ∠ 5) = 180 °$ .

binomial

A polynomial with two terms.

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.

center of dilation

See dilation.

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: $⨀ D$

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.

closure

A set is closed (under an operation) if and only if the operation on any two elements of the set produces another element of the same set.

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 $90 °$ .

completing the square

Completing the Square changes the form of a quadratic function from standard form to vertex form. It can be used for solving a quadratic equation and is one method for deriving the quadratic formula.

$\begin{matrix} standard form & vertex form \\ a x^{2} + b x + c = 0 \Rightarrow & a {(x + d)}^{2} + e \end{matrix}$

$if a = 1,$

$x^{2} + b x + c = 0 \Rightarrow {(x + \frac{b}{2})}^{2} + [c - {(\frac{b}{2})}^{2}]$

$vertex (- \frac{b}{2}, [c - {(\frac{b}{2})}^{2}])$

$Example: \frac{6}{2} = 3 5 - 3^{2}$

$x^{2} + 6 x + 5 = 0 \Rightarrow {(x + 3)}^{2} + (- 4)$

$vertex (- 3, - 4)$

complex conjugates

A pair of complex numbers whose product is a nonzero real number.

The complex numbers $(a + b i)$ and $(a - b i)$ form a conjugate pair.

The product $(a + b i) (a - b i) = a^{2} + b^{2}$ , a real number.

The conjugate of a complex number $a + b i$ is the complex number $a - b i$ .

The conjugate of a complex number is represented with the notation $\overset{―}{a + b i}$ .

complex number

A number with a real part and an imaginary part. A complex number can be written in the form $a + b i$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

When $a = 0$ , the complex number $0 + b i$ can be written simply as $b i .$ It is then referred to as a pure imaginary number.

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.

conditional frequency

See two-way relative frequency table.

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.

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.

constant of proportionality/ constant of variation

The constant of proportionality (also called constant of variation) is encountered in direct variation and inverse variation equations.

It is usually symbolized by $k$ .

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:

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).

cube root

A value that, when multiplied by itself, three times gives the number.

Example:

$3 \times 3 \times 3 = 27$ , so the cube root of $27$ is $3$ or $\sqrt[3]{27} = 3$ .

$4 \times 4 \times 4 = 64$ , so the cube root of $64$ is $4$ or $\sqrt[3]{64} = 4$ .

The mathematical symbol that indicates to find the cube root is a radical sign with a small $3$ on the outside. $\sqrt[3]{}$ .

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 $60 °$ .

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 $\frac{O P^{^{'}}}{O P} = k$ .

direct variation

disjoint

See mutually exclusive.

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.

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.

extraneous solution

A derived solution to an equation that is invalid in the original equation.

factored form of a quadratic function

Go to quadratic function.

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.

Fundamental Theorem of Algebra

An $n th$ degree polynomial function has $n$ roots, but some of those roots might be complex numbers.

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 $\frac{a}{b} = \frac{b}{c}$ .

hexagon

A six-sided polygon.

horizontal asymptote

A line that the graph approaches but does not reach. Exponential functions have a horizontal asymptote. The horizontal asymptote is the value the function approaches as $x$ either gets infinitely larger or smaller. An asymptote is an imaginary line, but it is often shown as a dotted line on the graph.

$f (x) = 2^{x}$

As $x$ gets smaller, the graph of $f (x)$ approaches the horizontal asymptote, $y = 0$ .

$h (x) = 2^{- x}$

As $x$ gets larger, the graph of $h (x)$ approaches the horizontal asymptote, $y = 0$ .

$g (x) = 2^{x} - 3$

As $x$ gets smaller, the graph of $g (x)$ approaches the horizontal asymptote, $y = - 3$ .

M–R

marginal frequency

See two-way tables.

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.

number sets (systems)

Your first experience with number sets was probably when you learned to count. This set is called the Natural numbers, $N = {1, 2, 3, \dots}$ . When you added $0$ the set grew to be the Whole numbers, $W = {0, 1, 2, 3, \dots}$ . The need for Integers, $Z = {\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots}$ , arose when you subtracted a large number from a smaller number. Then you needed the Rational numbers, $Q = {\frac{i n t e g e r}{i n t e g e r}}$ , when you started dividing. Other number sets (or systems) are needed in more advanced mathematics.

octagon

An eight-sided polygon.

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

The graph of every equation that can be written in the form $y = A x^{2} + B x + C$ , where $A \geq 1$ is in the shape of a parabola. It looks a bit like a U but it has a very specific shape. Moving from the vertex, it is the exact same shape on the left as it is on the right. (It is symmetric.)

The graph of the parent function or

$y = x^{2}$ follows the pattern:

move right 1 step, move up $1^{2}$ or $1$
move right 2 steps, move up $2^{2}$ or $4$
move right 3 steps, move up $3^{2}$ or $9$

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.

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.

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