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.

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

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.

quadratic formula

The quadratic formula allows us to solve any quadratic equation that’s in the form $a x^{2} + b x + c = 0$ . The letters $a$ , $b$ , and $c$ in the formula represent the coefficients of the terms.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

quadratic function

quadratic inequality

A function whose degree is $2$ and where the $y$ is not always exactly equal to the function. These types of functions use symbols called inequality symbols that include the symbols we know as less than $<$ , greater than $>$ , less than or equal to $\leq$ , and greater than or equal to $\geq$ .

Example: $y \geq a x^{2} + b x + c$

quadrilaterals: types

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

radical

A radical is the mathematical inverse of an exponent. This is the symbol for a radical: $\sqrt{x}$ . It is also called a square root symbol, but that is only when it’s asking for the number that when multiplied by itself gives you the number inside the $\sqrt[2]{x}$ . (The $2$ is not usually written.) It can be used to indicate a cube root $\sqrt[3]{x}$ , a fourth root $\sqrt[4]{x}$ , or higher. (A root that is higher than $2$ is written in.)

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.

rational exponent (fractional exponent)

Rational exponents (also called fractional exponents) are expressions with exponents that are rational numbers (as opposed to integers).

$a^{\frac{1}{n}} = \sqrt[n]{a}$

$a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} or \sqrt[n]{(} a)^{m}$

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.

relative frequency table (statistics)

When the data in a two-way table is written as percentages

See two-way frequency table.

rhombus

A quadrilateral in which all sides are congruent.

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.

roots: real and imaginary

The solutions of an equation in the form $f (x) = 0$ .

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.

scale factor

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

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.

special right triangles

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

square

See quadrilaterals: types.

square root

The square root of a number is one of the two identical factors that when multiplied together equal the number.

Example: $4 \times 4 = 1$ 6, so a square root of $16$ is $4$ .

Note that $(- 4) \times (- 4) = 16$ too. That means $- 4$ is also a square root of $16$ . The mathematical symbol that indicates to find the square root is a radical sign $\sqrt{}$ .

square root function

A function that has a radical (square root sign) and the independent variable $x$ is under the square root sign $y = \sqrt{x}$ .

Equation: $y = a \sqrt{x - h} + k$

Domain: ${x \in R | x \geq 0}$

Range: ${y \in R | y \geq 0}$

Always increasing

SSS triangle similarity

See triangle similarity.

standard form of a quadratic function

$a x^{2} + b x + c, a \neq 0$

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.

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.

transformations on a function (non-rigid)

A dilation is a nonrigid transformation $k f (x)$ because the shape changes in size. It will make the function change faster or slower depending on the value of $k$ . If $k > 1$ , it will grow faster and look like it has been stretched. If $0 < k < 1$ , the function will change more slowly and will appear to be fatter. A dilation is also called a vertical stretch.

transformations on a function (rigid)

A shift up, down, left, or right, or a vertical or horizontal reflection on the graph of a function is called a rigid transformation.

Vertical shift

$f (x) + k$

Up when $k > 0$

Down when $k < 0$

Horizontal shift

$f (x + k)$

Left when $k > 0$

Right when $k < 0$

Reflection

$- f (x)$ reflection over the $x$ -axis

$f (- x)$ reflection over the $y$ -axis

A dilation is a nonrigid transformation. It will make the function changes faster or slower depending on the value of $k f (x)$ . If $k > 1$ , it will grow faster and look like it has been stretched. If $0 < k < 1$ , the function will grow more slowly and will appear to be fatter.

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.

trinomial

A polynomial with three terms.

two-column proof

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

two-way frequency and two-way relative frequency table

A two-way frequency chart simply lists the number of each occurrence.

	Average is more than 100 texts sent per day	Average is less than 100 texts sent per day	Total
# of Teenagers	20	4	24
# of Adults	2	22	24
Totals	22	26	48

In a two-way relative frequency table, each number in the cells is divided by the grand total. That is because we are looking for a percentage that shows us how the data compares to the grand total.

	Average is more than 100 texts sent per day	Average is less than 100 texts sent per day	Total
% of Teenagers	42%	8%	50%
% of Adults	4%	46%	50%
% of Total	46%	54%	100%

In this table, the ‘inner’ values represent a percent and are called conditional frequencies. The conditional values in a relative frequency table can be calculated as percentages of one of the following:

the whole table (relative frequency of table)
the rows (relative frequency of rows)
the columns (relative frequency of column)

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

vertex

See angle.

vertex form

See quadratic function.

vertical angles

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

vertical asymptote

See asymptote.

vertical shift

See transformations on a function (rigid).

vertical stretch

See transformations on a function (non-rigid).

x-intercept

The point(s) where a line or a curve cross the $x$ -axis. The $y$ -value of the point will be $0$ . A non-horizontal line will only cross the $x$ -axis once. A curve could cross the $x$ -axis several times.

Y–Z

zeros, roots, solutions: The real solutions to a quadratic equation are where it is equal to zero. They are also called zeros or roots. Real zeros correspond to the $x$ -intercepts of the graph of a function.