Geometry

Truth Table

A diagram used to show the truth value of a compound statement for all possible combinations of truth values of its basic statements.

Sphere

A three-dimensional figure shaped like a circular ball. The volume of a sphere is V= 4 pi r^3/3 where r is the radius.

polyhedron

A three-dimensional figure whose faces are all polygons.

Prism

A three-dimensional figure with one lateral area and two bases. The type of prism is determined by which polygon forms its base. Several prisms are shown below.

Compass

A tool used to draw circles and arcs

Coplanar

Lying in the same plane.

Adjacent

Next to, such as adjacent sides in a polygon. In the diagram of polygon ABCDE, adjacent sides and are marked in red.

edge

Where two faces of a polyhedron meet.

Rectangular Solid

A 3-dimensional figure with rectangular bases and sides; a rectangular prism; a box.

Octagon

An 8-sided polygon.

Equidistant

Being the same distance apart.

Congruent

Coinciding; having the same measure or size. •Congruent angles have the same measure. •Congruent line segments have the same length. •Congruent triangles coincide with one is superimposed over the other. They have the same shape and size. •Congruence can be extended to other geometric shapes.

CPCTC

Corresponding parts of congruent triangles are congruent. This applies both to angles and sides.

Similar Figures

Figures of the same shape, but differing in size. Corresponding or "matching" angles are congruent; corresponding or "matching" sides are proportional in length. The diagram below shows similar figures.

Discriminant

For a quadratic equation of the form ax^2 + bx + c = 0, a ≠ 0, the discriminant is b^2 − 4ac. The nature of the discriminant gives important information about the roots of the quadratic equation: •If b2 − 4ac > 0, then the quadratic has two, real roots. If, in addition, b2 − 4ac is a perfect square, the roots are rational; otherwise they are irrational. •If b2 − 4ac = 0, then the quadratic has one, real (double) root. •If b2 − 4ac < 0, then there are no real roots; the roots of the quadratic are imaginary.

Ratio

For real numbers a and b where b ≠ 0, the ratio of a to b is the fraction a/b or . A ratio can also be written as a:b.

DeMorgan's Law

Gives equivalent expressions for the negation of a conjunction or of a disjunction. The negation of the conjunction "a and b" is the disjunction "not a or not b." The negation of the disjunction "a or b" is the conjunction "not a and not b." In symbolic form: To negate a conjunction, negate the original conjuncts and join them in a disjunction: ~(a b) ↔ (~a ~b). To negate a disjunction, negate the original disjuncts and join them in a conjunction: ~(a b) ↔ (~a ~b).

Transitive Postulate

If a = b and b = c, then a = c. If a < b and b < c, then a < c.

Midpoint Formula

In the coordinate plane, the coordinates of the midpoint of the line segment whose endpoints are (x1, y1) and (x2, y2) can be calculated as: (x,y) (x1+x2 /2 , y1+y2 /2)

Distance Equation

In the coordinate plane, the distance, d, between the points (x1, y1) and (x2, y2) can be calculated as: d = square root (x2-x1)^2 + (y2-y1)^2 •This equation can also be used to find the length of a line segment with endpoints (x1, y1) and (x2, y2).

Factor

One of the multiplicands in a product. For example, in the product, 6x, the factors are the constant 6 and the variable x; in the product (x + 2)(x - 1), the factors are the binomials (x + 2) and (x - 1).

Parallel Planes

Planes that do not intersect. In the diagram below, plane P is parallel to plane Q.

Invariant

Unchanged after a transformation.

Reciprocal

For a real number a ≠ 0, 1/a or is its reciprocal or multiplicative inverse. •The product of a number and its reciprocal is always equal to 1. For example, the reciprocal of 6 is 1/6. •For a fraction a/b where a ≠ 0, the reciprocal is the fraction b/a. For example, the reciprocal of −2/3 is −3/2.

Absolute Value

For any real number a, the absolute value of a is denoted by |a|. |a| = a if a ≥ 0; otherwise, |a| = −a. •On the real number line, |a| is the distance between the coordinate a and the origin. •For example, |2| = 2, and |−5| = 5.

Power

For any real number b and positive integer n, b raised to the nINth power is defined as: b^n = b*b***b/n for a total of n factors of b. The number b is called the base and n is called the exponent. Here are some rules for working with powers: •For n = 0 and b ≠ 0: b^0 = 1. •For a negative integer −n: image. •For a rational number 1/n, where n is a positive integer: b 1/n = n radical b which is the ninth-root of b. •For a rational number n/m , where n and m are positive integers: b m/n = (n radical b) m.

Base of a Power

For any real number b and positive integer n, b raised to the ninth power is b^n=b*b***b/n for a total of n factors of b. •b is called the base •n is called the exponent. •For example, 2^3 = 2×2×2 = 8. •b0, meaning zero factors of b, is 1, as long as b ≠ 0; and 0n = 0.

Exponent

For any real number b and positive integer n, b raised to the ninth power is defined as: b^n = b*b***b/n for a total of n factors of b. b is called the base and n is called the exponent. Here are some rules for working with exponents: •For n = 0 and b ≠ 0: b^0 = 1. •For a negative integer −n: . •For a rational number 1/n, where n is a positive integer: b 1/n = n radical b which is the ninth-root of b. •For a rational number n/m, where n and m are positive integers: b m/n = (n radical b)^m

Logarithm

For b > 1, if y = bx, then logb y = x. That is, y = logb x is the inverse function of y = bx. Here are some rules for working with logarithms when m and n are positive real numbers: •logb mn = logb m + logb n •logb (m/n) = logb m − logb n •logb nk = k logb n •logb b = 1

Disjunctive Addition, Law of

Given p, then the disjunction p V q is true regardless of whether q is true or false.

Modus Tollens, Law of

Given that a conditional statement is true and that its consequent is false, its antecedent must also be false. In symbolic form: [(a → b) ^ ~b] → ~a

Disjunctive Inference, Law of

Given that a disjunction is true, and that one of its disjuncts is false, then the other disjunct must be true. In symbolic form: [(a V b) ^ ~a] → b

ray

Half of a line, starting with one endpoint and extending indefinitely in one direction. The ray with endpoint A and another point B can be named as ray AB

Addition Postulate

If equal quantities are added to equal quantities, the sums are also equal.

Subtraction Postulate

If equal quantities are subtracted from equal quantities, the differences are also equal.

Composition of Functions

If f(x) and g(x) are functions, then the composition of f and g, denoted by f ° g, is defined as f(g(x)) for all values of x in the domain of g for which f is defined. •The result of a composition of functions is found by working from right to left. •For example, in f ° g, first apply the rule for g, then apply the rule for f on that result. •For example, if f(x) = 2x + 1 and g(x) = 4x, then f ° g(x) = f(g(x)) = 2(4x) + 1 = 8x+ 1.

Range (of a Relation)

In a relation, the set of all second members of the ordered pairs in the relation.

Leg

In a right triangle, a side that is not opposite the right angle. •In the diagram of a right triangle, the legs are shown in red. •The side opposite the right angle is called the hypotenuse.

Cosine Ratio (cos)

In a right triangle, the cosine of either of the acute angles is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.

Sine Ratio (sin)

In a right triangle, the sine of either of the acute angles is the ratio of the length of the leg opposite the angle to the length of the hypotenuse.

Tangent Ratio (tan)

In a right triangle, the tangent of either acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

Image

In a transformation in which point A is mapped onto point A', point A' is called the image of point A. •An entire figure can be mapped onto a corresponding image under a transformation. •Translation, dilation, reflection in a line, and rotation are types of transformations.

Substitution Postulate

In any expression, two quantities that are equal may be substituted for one another.

Origin

The origin of a number line is the zero point, shown in red in the diagram below: The origin of the coordinate plane is the point corresponding to the coordinates (0, 0), as shown in red in the diagram below. It is the point of intersection of the x- and y-axes.

sector of a circle

The part of a circle enclosed by two radii and the arc they intercept. A pie-shaped piece of a circle.

Disjunct

The parts of a disjunction. In the disjunction, "a or b," written symbolically as a V b, a and b are the disjuncts.

Percentage

The percentage x, written as x%, is equivalent to the fraction x/100. The phrase "x% of the number n" can be expressed as the product: x/100 x n.

Circumference

The perimeter of a circle, or the distance measured around the circle. For a circle of radius r, the circumference is 2πr.

Perpendicular Bisector

The perpendicular bisector of a line segment is a line that is perpendicular to the given line segment and also divides the segment into two segments of the same length. In the diagram below, line CD is a perpendicular bisector of line AB .

Coordinate Plane

The plane formed by the intersection at right angles of a horizontal real number line (the x-axis) and a vertical real number line (the y-axis). The point of intersection, called the origin, is at the 0-coordinate of the two number lines. The diagram shows the xy-coordinate plane.

Break-even Point

The point at which the income a business is making is equal to its expenses; at this point, the business is neither making money nor losing money.

Orthocenter

The point at which the lines containing the altitudes of a triangle are concurrent, that is, where the altitudes of a triangle intersect. In the diagram, the orthocenter of the triangle is marked in red.

Triangle Inequality

The length of any side of a triangle is less than the sum of the lengths of the other two sides.

x-Coordinate

The location along the x-axis of a point in the coordinate plane. •All points in the coordinate plane can be uniquely determined by an ordered pair of real numbers (a, b), where a is the x-coordinate. •For example, the x-coordinate of the point in red on the graph is 3.

y-Coordinate

The location along the y-axis of a point in the coordinate plane. •All points in the coordinate plane can be uniquely determined by an ordered pair of real numbers (a, b), where b is the y-coordinate. •For example, the y-coordinate of the point in red on the graph is 2.

Radian

The measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. For example, the diagram shows a circle of radius r centered at the origin. The angle θ is shown in standard position. The length of the arc cut by angle θ is r, and the measure of θ is 1 radian.

Midpoint

The midpoint, C, of line segment AB is a point lying on segment AB that divides into two segments of equal length, that is, AC = CB.

Negation

The opposite or negation of statement a is expressed as "not a." Its truth value is the opposite of the truth value of the original statement. For example, consider the statement "It is sunny." Its negation is "It is not sunny." In symbolic form, the negation of a is expressed as ~a. The table below shows the relationship between a statement and its negation.

Orientation

The order of the vertices in a polygon. Listing the vertices from one vertex to the next adjacent vertex is either clockwise orientation or counter-clockwise orientation.

Undefined

The value of some expressions is undefined under certain conditions. For example, the value of the fraction a/b is undefined if b = 0; the slope of a vertical line is undefined.

Root of an Equation

The value or values that a variable may have which makes an equation a true statement. For example, for the equation 2x 4 = 0, the value x = 2 makes the equation a true statement; thus 2 is a root of the equation. The roots of the equation x^2 = 9 are 3 and -3.

Integers

The whole numbers, their additive opposites, and zero: ... , −3, −2, −1, 0, 1, 2, 3, ...

Reflexive Property

These facts illustrate the reflexive property: •Every real number is equal to itself, that is, for all real numbers a, a = a. •Any logical statement p is equivalent to itself, that is, p ↔ p. •Any angle is congruent to itself, that is, ∠A ≅ ∠A.

Concurrent Lines

Three or more lines that contain the same point, that is, they all intersect at the same point. In the diagram, lines p, q and r are concurrent. The point of concurrency is S, marked in red.

bisect

To divide into two equal portions.

inscribe

To draw one figure inside another figure so that every vertex of the inner figure touches the outer figure.

Circumscribe

To draw one figure outside another figure so that every vertex of the inner figure touches the outer figure.

Similar Triangles

Triangles of the same shape but differing in size. •Corresponding or "matching" angles are congruent. •Corresponding or "matching" sides are proportional in length. •Two triangles having two sets of corresponding angles with equal measure are similar. •"Triangle A is similar to triangle B" is written as: ΔA ~ ΔB.

Complementary Angles

Two angles are complementary if their measures add up to 90°. In the diagram, ∠ADB and ∠BDC are complementary.

Supplementary Angles

Two angles are supplementary if the sum of their measures equals 180°. In the diagram below, ∠1 and ∠2 are supplementary

Adjacent Angles

Two angles that share a common vertex and a common ray (side), but that do not share any interior points. •For example, in the diagram below, ∠BAC and ∠CAD share vertex A and side , but no interior points; they form a pair of adjacent angles. •∠BAC and ∠BAD share vertex A and side , but they are not adjacent angles because the points interior to∠BAC are also interior to ∠BAD.

skew lines

Two lines that do not lie in the same plane and that do not intersect. In the diagram, plane P is parallel to plane Q. Lines m and n are skew lines

Angle

Two rays (half-lines) sharing a common endpoint (vertex) form an angle. •The angle at vertex A (shown below, left, in red) is referred to as ∠A. •When a diagram contains more than one angle, the angle is referred to by the name of a point on one ray, followed by the vertex, followed by a point on the second ray. •For example, the angle shown below, right, in red, is referred to as ∠QPR.

Congruent Triangles

Two triangles with the same size and shape. "Triangle A is congruent to triangle B" is expressed as ΔA ≅ ΔB. ΔA and ΔB are congruent if any of the following conditions is true: •The sides of ΔA are congruent to the corresponding sides of ΔB. This condition is called SSS (side-side-side) •Two sides and the included angle in ΔA are congruent to the corresponding portions of ΔB. This condition is called SAS (side-angle-side) •Two angles and the included side in ΔA are congruent to the corresponding portions of ΔB. This condition is called ASA (angle-side-angle) •Two angles and the side opposite to one of these angles in ΔA are congruent to the corresponding portions of ΔB. This condition is called AAS (angle-angle-side)

Logical Connectives

Used to form compound statements, including: •the conjunction "and" ( ˄ ) •the disjunction "or" ( V ) •the conditional "if...then" (→) •the biconditional "if and only if" (↔).

Point Reflection

When a point P is reflected in a point O, point O is the midpoint of the line segment joining P and its image, P'.

Corresponding Angles

When a transversal intersects a pair of parallel lines, corresponding angles lie on the same side of the transversal. •One is an interior angle and the other is an exterior angle. •Corresponding angles have the same measure. •In the diagram to the left, parallel lines l and m are crossed by transversal t; a pair of corresponding angles is shown in red. In a pair of similar or congruent polygons, corresponding angles "match" and thus have the same measure.

Pentagon

A 5-sided polygon. •The sum of the measures of the interior angles of a pentagon is 540°. •The diagram shows a non-regular pentagon, that is, its sides are of unequal length.

Hexagon

A 6-sided polygon. The diagram shows a regular hexagon

Symmetry

A balanced arrangement about a line or a point.

Associative Property

A binary operation @ is associative if (a @ b) @ c = a @ (b @ c). Addition and multiplication of real numbers are associative binary operations: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) but, in general, subtraction and division are not: (a − b) − c ≠ a − (b − c) and (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Diameter

A chord that passes through the center of a circle. The length of a diameter is twice the length of a radius of the circle. The diagram shows diameter line AB of circle O.

Circle

A circle consists of all points in a plane that are a fixed distance r from a given central point. •The distance, r, is called the radius. •The fixed central point is called the center. •A circle is often referred to by its center point. •In the diagram, circle O has point O as its center and r as its radius. •The circumference (perimeter) of circle O is 2πr. •The area of circle O is πr2. •In the coordinate plane, the general equation of a circle of radius r centered at point O(h, k) is (x − h)2 + (y − k)2 = r^2.

Center of a Circle

A circle consists of all points in a plane that are a fixed distance r, called the radius, from a given central point, called the center. •A circle is often named after its center. •In the diagram, circle O has point O as its center and r as its radius. •The circumference of circle O is 2πr. The area of circle O is πr2. •In the coordinate plane, the general equation of a circle of radius r centered at point O(a, b) is (x − a)2 + (y − b)2 = r^2.

Circumscribed Circle

A circle drawn around a triangle or other polygon that passes through each of its vertices. In the diagram, the circle, in blue, is circumscribed about a triangle.

Inscribed Circle

A circle drawn inside a triangle or other polygon so that each of the polygon's sides is a tangent of the circle. The diagram below shows a circle, in blue, inscribed inside a triangle

Incircle

A circle inscribed in a triangle. It is tangent to all three sides of the triangle. In the diagram of the triangle, the incircle is shown in blue.

Data Set

A collection of data items; for example, a set of test scores, the coins in your pocket, the results of a survey. •Data is often organized into tables or graphs for easier viewing. •Measures of central tendency (mean, median, mode, standard deviation) can be calculated for sets of numeric data items.

Converse

A compound statement formed by interchanging the antecedent and consequent of a conditional statement. For example, consider the conditional statement "If it is sunny, then we will go on a picnic." Its converse is "If we go on a picnic, then it is sunny." In symbolic form, the converse of a → b is b → a. The truth table below shows the relationship between a conditional statement and its converse.

Inverse (logical)

A compound statement formed by negating both the antecedent (the "if" part) and consequent (the "then" part) of a conditional statement. The inverse of the conditional statement "if a then b" is "if not a then not b." For example, consider the conditional statement, "If it is sunny, we will go on a picnic." The inverse of this statement is "If it is not sunny, we will not go on a picnic." In symbolic form, the inverse of the conditional statement a → b is ~a → ~b. The truth table below shows the relationship between a conditional statement and its inverse.

Variable

A letter used to represent a real number; x, y, and n are commonly used variables.

Tangent to a Circle

A line or line segment that intersects a circle in exactly one point. In the diagram of circle O, is a tangent

Secant of a Circle

A line or line segment that intersects a circle in exactly two points. In the diagram of circle O, image is a secant.

Altitude

A line segment drawn from one point or side in a plane figure that meets another side at a right angle. •Altitudes are often constructed in triangles, parallelograms, and trapezoids, because the length of the altitude (height) is used in computing the area enclosed by the figure. •In the figures below, altitudes are shown in red.

Median of a Triangle

A line segment drawn from one vertex of a triangle to the midpoint of the opposite side. In the figure below, line BD is a median of ΔABC.

midsegment

A line segment joining the midpoints of two sides of a triangle. •The midsegment is parallel to the third side of the triangle. •The length of the midsegment is one-half the length of the third side of the triangle.

Median of a trapezoid

A line segment that connects the midpoints of the two legs of a trapezoid. It is parallel to the bases.

SAS

A theorem stating a condition for the congruence of two triangles. ΔA and ΔB are congruent if two sides and the included angle in ΔA are congruent to the corresponding portions of ΔB.

AAA

A theorem stating a condition for the similarity of two triangles. ΔA and ΔB are similar if the angles of ΔA are congruent to (have the same measure as) the corresponding angles of ΔB.

Vertical Shift

Adding a constant to a function shifts its graph vertically. If a vertical translation is written f(x) + d, then a positive value for 'd' produces an upward shift and a negative value for 'd' produces a downward shift.

Horizontal Shift

Adding a constant to the x-value of a function shifts its graph horizontally. For example, if k is positive, then f(x + k) will produce a horizontal shift to the left 'k' units, whereas f(x - k) will produce a horizontal shift to the right 'k' units.

Detachment, Law of

Also known as Modus Ponens; if a conditional statement and its antecedent are true, the consequent must also be true. That is, if a is true and a implies b, then b is also true. In symbolic form: [a ^ (a → b)] → b

Syllogism

Also known as the Chain Rule; If the conditional statements "if a then b" and "if b then c" are both true, then the conditional statement "if a then c" is also true. In symbolic form: [(a → b) ˄ (b → c)] → (a → c).

Modus Ponens, Law of

Also known as the Law of Detachment; given that a conditional statement and its antecedent are true, the consequent must also be true. In symbolic form: [a ^ (a → b)] → b

Chain Rule

Also known as the Law of Syllogism. If the conditional statements "if a then b" and "if b then c" are both true, then the conditional statement "if a then c" is also true. In symbolic form: [(a → b) (b → c)] → (a → c).

FOIL

An acronym (First terms, Outer terms, Inner terms, Last terms) referring to a method for expanding the product of two binomials (ax + b) and (cx + d) by applying the associative and distributive properties: (ax+b) X (cx + d) = acx^2 + adx + bcx + bd F O I L = acx^2 + (ad + bc ) x + bd

Polynomial

An algebraic expression consisting of the sum of individual monomial terms. For example: 4x3 7x2 + 2x 1.

Interior Angle

An angle formed by adjacent sides of a polygon, interior to the polygon. In the diagram of ΔABC, ∠A, ∠B and ∠ACB are interior angles, while ∠ACD is an exterior angle.

Right Angle

An angle measuring 90°. Geometric diagrams often use a small square to show that two lines or line segments meet at a right angle:

Straight Angle

An angle whose measure is 180°. For example, in the diagram below, ∠ABC is a straight angle.

Acute Angle

An angle whose measure is between 0° and 90°. The diagram to the left shows an acute angle.

Obtuse Angle

An angle whose measure is between 90° and 180°. The diagram below shows an obtuse angle.

Conjunction

A compound statement of the form "a and b." a and b are called the conjuncts. The conjunction "a and b" is true only if both a and b are true. The statement "Bob studies and Bob passes the math test" is an example of a comjuntion; the conjuncts are "Bob studies" and "Bob passes the math test." In symbolic form, the conjunction "a and b" is expressed symbolically as a ^ b. The truth table below shows the truth value of a ^b.

Biconditional (if and only if )

A compound statement of the form "a if and only if b." If a and b have the same truth value, then the biconditional statement "a if and only if b" is true; if a and b do not have the same truth value, then "a if and only if b" is false. The statement "Bob passes math if and only if Bob studies" is an example of a biconditional statement. In symbolic form, the biconditional statement "a if and only if b" is expressed as a ↔ b. The truth table below shows the truth value of a ↔ b.

Disjunction

A compound statement of the form "a or b." The statements a and b are called the disjuncts. The disjunction "a or b" is true if either a or b (or both) are true. The statement "It is sunny or we are going on a picnic" is an example of a conjuntion; the conjuncts are "It is sunny" and "We are going on a picnic." In symbolic form, the disjunction "a or b" is expressed symbolically as a V b. The truth table below shows the truth value of a V b.

Conditional Statement

A compound statement of the form "if a then b". The statement a is called the antecedent or hypothesis. The statement b is called the consequent or conclusion. The conditional statement "if a then b" is always true unless a is true and b is false. The statement "If Bob studies, then he will pass the math test" is an example of a conditional statement. The antecedent is "Bob studies" and the consequent is "Bob passes the math test." In symbolic form, the conditional statement is expressed symbolically as a → b. The truth table below shows the truth value of a → b.

Contradiction

A compound statement that is always false. For example, the statement "a and not a" is a contradiction, because a logical statement cannot be both true and false at the same time.

Contrapositive Inference, Law of

A conditional statement "if a then b" and its contrapositive "if not b then not a" are logically equivalent. In symbolic form: (a → b) ↔ (~b → ~a)

Coefficient

A constant factor in a monomial expression. For example, 3 is the coefficient of 3x; −7 is the coefficient of −7x^3y.

Tree Diagram

A diagram with branches used to describe the possible outcomes in the steps of a probability experiment. Each branch is labeled with probability of the outcome. This makes it easy to compute the probability of combined events. Consider this example of a two-step, unequal chance problem, without replacement. A cooler contains 6 sodas; 3 are orange (O), 2 are colas (C), and 1 is a root beer (RB ). Pete chooses a soda from the cooler without looking, and does not replace it. Then Patty chooses a soda from the 5 that are left in the cooler. The tree diagram below illustrates the experiment. Each branch represents one soda. The first set of branches demonstrates Pete's choice; the second set shows Patty's choices. Pete has an unequal chance of getting any of the three flavors. Which sodas are left for Patty to choose from depends on what Pete took.

Scale Drawing

A drawing that is proportional to an actual object

Constant

A fixed number, such as the integer, 5 or the decimal 3.3, or the fraction 1/2 , or the irrational number π.

Quadrilateral

A four-sided polygon. Several familiar kinds of quadrilaterals are shown below.

Exponential Function

A function of the form y = b^x where the base b > 1. The diagram shows the graph of the exponential function y = 2^x. As x decreases in value, the graph of the function approaches the x-axis as an asymptote.

Radius

A line segment whose endpoints are the center of a circle and a point lying on the circle. •In the diagram, circle O has point O as its center and r as its radius. •The circumference of circle O is 2πr. •The area of circle O is πr2. •In the coordinate plane, the general equation of a circle of radius r centered at point O(a, b) is (x − a)^2 + (y − b)^2 = r^2.

Diagonal

A line segment whose endpoints are two non-adjacent vertices of a polygon. For example, in the diagram of a parallelogram below, the diagonals, shown in red, connect opposite vertices.

Chord

A line segment whose endpoints lie on a circle. In the diagram, line AB is a chord of circle O.

Transversal

A line that intersects another pair of lines in two distinct points. For example, in the diagram below, is a transversal intersecting parallel lines and at points G and H, respectively. •When a transversal crosses a pair of parallel lines, all of the angles formed are either congruent or supplementary.

Asympote

A line that is approached, although never intersected, by a graph as x increases or decreases. •For example, in the graph below, left, of the equation y = 3x, as x decreases, the graph approaches the x-axis as a horizontal asymptote. •The graph below, right, of the equation xy = k is restricted to Quadrants II and IV. In Quadrant II, as x decreases, the graph approaches the x-axis as a horizontal asymptote; as x increases towards 0, the graph approaches the y-axis as a vertical asymptote. In Quadrant IV, as x increases, the graph approaches the x-axis as a horizontal asymptote; as x decreases towards 0, the graph approaches the y-axis as a vertical asymptote.

Bisector of a Segment

A line, or portion of a line, that divides a given line segment into two segments of equal length. The bisector of a segment contains the midpoint of the segment. In the diagram below, the perpendicular bisector of line AB is shown in red.

Bisector of an Angle

A line, or portion of a line, that divides an angle into two angles of equal measure. In the diagram below, the bisector of ∠A is shown in red.

Slope-Intercept Form

A linear equation of the form y = mx + b. •The slope of the line is m. •Its graph intersects the y-axis at the point (0, b). •b is called the y-intercept.

point

A location; a point has no dimensions. A point is usually referred to by a letter name and represented by a dot.

Proof

A logical sequence of steps that shows a set of given assumptions leads to a particular conclusion.

Transformation

A mapping of one set of points into another set of points, called the image, according to a rule of some kind. •An entire figure can be mapped into a corresponding image under a transformation. •Translation, dilation, reflection in a line, and rotation are types of transformations.

Equation

A mathematical statement equating the values of two algebraic expressions. For example: the equation 3x − 4 = 5 says that the value of the algebraic expression 3x − 4 is equal to the constant 5.

Inequality

A mathematical statement relating the values of two algebraic expressions. •The strict comparisons include "less than" which uses the symbol <; and "greater than," which whish uses the symbol > . •The non-strict comparisons are "less than or equal to," which uses the symbol ≤; and "greater than or equal to," which uses the symbol ≥. •For example: the inequality 3x − 4 > 5 expresses the fact that the value of the algebraic expression 3x − 4 is greater than the constant 5.

height

A measure of how tall a figure or object is.

Solution

A possible value of a variable that makes a statement about the variable true. For example, one possible solution of the equation x^2 = 9 is −3; another possible solution is 3.

Area

A measure of the amount of space enclosed by a plane figure. Here is how to calculate the area of some common figures: •The area of a square with side of length s is s2. •The area of a rectangle of length l and width w is lw. •The area of a parallelogram with base b and height h is bh. •The area of a triangle with base b and height h is . •The area of a circle of radius r is πr2.

Volume

A measure of the amount of space enclosed by a solid. Here is how to compute the volume of some common solids: •For a rectangular prism with length l, width w and depth d, V = l×w×d. •For a cube with side of length s, V = s^3. •For a right cylinder with base of radius r and height h, V = πr^2h. •For a right cone with base of radius r and height h, . V=1/3πr^2h

Slope

A measure of the steepness of a line. If two points (x1, y1) and (x2, y2) lie on line L and x1 ≠ x2, the slope, m, of line L is given by: . •If m > 0, the graph of L slants upward as the value of x increases. •If m < 0, the graph of L slants downward as x increases. •The greater the absolute value of m ( |m| ), the steeper the line. •The slope of a horizontal line is 0. •The slope of a vertical line is undefined. •For a linear equation of the form y = mx + b, the slope of the line is m, and its graph intersects the y-axis at the point (0, b).

Decimal Notation

A method of representing a real number as the sum of both positive and negative integer powers of 10. In decimal notation, the coefficients of the powers of 10 are always integers between 0 and 9. For example, the decimal 123.456 literally means: 123.456=(1*10^2)+(2*10^1)+(3 *10^0)+(4*10^-1)+(5*10^-2)+ (6*10^-3) •Digits to the left of the decimal point indicate coefficients of nonnegative powers of 10, with exponents beginning with 0 and increasing from right to left. •Digits to the right of the decimal point indicate coefficients of negative powers of 10, decreasing from left to right. •If the decimal point is omitted, it is assumed to be immediately to the right of the rightmost digit.

Indirect Method

A method often used to prove that a conditional statement is true. The proof begins by assuming that the consequent is false, and proceeds to show, by the application of the rules of logic, that this leads to a contradiction.

Algebraic Expression

A notation used to indicate an operation or series of operations involving real numbers. Variables are used to represent numbers. Various symbols are used to denote operations, such as + for addition, − for subtraction, × or · for multiplication, / or ÷ for division. Examples: x + y denotes the sum of x and y; 2b − a indicates that the number a is subtracted from twice the number b; x2 denotes the square of x, or x·x. Parentheses are sometimes used to indicate the order of evaluation. Evaluate expressions inside parentheses first. For example, x(y+ 1)2 means take the quantity y, add 1 to it, square that, then multiply by the quantity x.

Imaginary Number

A number of the form bi where b is a real number and i is the imaginary unit, i = square root of -1

linear pair of angles

A pair of angles that are both adjacent and supplementary. In the diagram, ∠ABD and ∠DBC are a linear pair of angles.

Perpendicular Lines

A pair of lines that intersect, forming a right angle, usually indicated by a small square, as shown in the diagram below. "Line AB is perpendicular to line CD" is expressed as: AB ⊥ CD.

Vertical Angles

A pair of nonadjacent angles formed by the intersection of two lines. Vertical angles have equal measure. The diagram below shows a pair of vertical angles.

Ordered Pair

A pair of numbers in which the order is significant. •In the ordered pair (a, b), the first member of the pair, a, is called the abscissa; the second member of the pair, b, is called the ordinate. •Points on the coordinate plane are denoted by an ordered pair of numbers. •If the horizontal axis is labeled x and the vertical axis is labeled y, then the ordered pair (a, b) corresponds to the point whose x-coordinate is a and whose y-coordinate is b.

Directrix of a Parabola

A parabola is the set of all points P in a plane that are equidistant from a fixed line and a fixed point not on the line. The fixed line is called the directrix of the parabola

Focus of a Parabola

A parabola is the set of all points P in a plane that are equidistant from a fixed line and a fixed point not on the line. The fixed point is called the focus.

Line Symmetry

A plane figure is symmetric about a given line l if the two halves of the figures are superimposed when the figure is "folded" along l. For example, the letter A shows vertical line symmetry; if you fold it along the vertical line shown in blue, below, the two halves are superimposed. The letter D shows horizontal line symmetry, shown in red. The letter H shows both vertical and horizontal line symmetry.

Point Symmetry

A plane figure is symmetric about a point, P, if any line drawn through P intersects the figure at equal distances and in opposite directions along the line. A simple test for point symmetry is to rotate the figure by 180° about P. If the image after rotation coincides with the original figure, then the figure is symmetric about the point P. For example, the letters O and Z are symmetric about their center points.

Point of Intersection

A point common to two or more lines, segments, or other geometric figures. The diagrams below show a circle and a line segment with 0, 1, and 2 points of intersection, respectively.

Regular Polygon

A polygon whose sides are all of the same length. •All of the interior angles of a regular polygon are congruent. •For a regular n-sided polygon with sides of length s, the sum of the measures of the interior angles is 180(n − 2), and the perimeter is n×s. •An equilateral triangle, a rhombus, a square, and a regular pentagon are all examples of regular polygons.

Triangle

A polygon with three sides. In ΔABC, the sides opposite ∠A, ∠B, and ∠C are commonly referred to as a, b, and c, respectively. The perimeter of a triangle is the sum of the lengths of its sides. For a triangle with base of length b and height h, the area is: A = 1/2 bh. Triangles have many special properties: •The sum of the measures of the three interior angles is always 180°. •The length of any side is less than the sum of the lengths of the other two sides; this is called the triangle inequality. •The longest side of a triangle is opposite the largest angle; the shortest side is opposite the smallest angle. •The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles.

Monomial

A polynomial having only one term; for example: x, 10, and −3x3y^2 are all monomials.

Binomial

A polynomial having two unlike terms; for example, x + 1, or 2x + 3, or 3x2 − 2x.

Trinomial

A polynomial with three terms. For example: 3x^2 + 5x − 7

Arc

A portion of a circle. The arc with endpoints A and B on a circle is referred to as: arc AB

Line Segment

A portion of a line that consists of two endpoints on the line, and all the points in between them. •A line segment is uniquely determined by its endpoints. •The line segment with endpoints A and B is called line AB . •Its length is called AB.

Probability

A probability experiment may have a number of outcomes. The probability of an event is the ratio of the number of ways in which the event can occur to the total number of outcomes. For example, in a simple coin toss, the outcomes are "heads" or "tails." If the coin is fair, either event is equally likely, that is, the probability of "heads" is 1/2, and the probability of "tails" is also 1/2.

Guess and Test

A problem solving strategy in which you make a good guess and check to see if it is correct. If not, make a better guess and keep checking and guessing until you come to the correct solution.

Distributive Property

A property linking two binary operations. For example, the distributive property of multiplication over addition is illustrated by: a(b + c) = ab + ac for real numbers a, b, and c.

Mean Proportional

A proportion in which the means are equal, for example: a/b = b/c.

Parallelogram

A quadrilateral (four-sided polygon) that has two pairs of parallel sides. •Opposite sides of a parallelogram are parallel and congruent. •Opposite angles are congruent. •Consecutive angles are supplementary. •In the diagram of parallelogram ABCD, line AB is parallel to line DC , line AD is parallel to line BC , AB = DC = b, and AD = BC = a. m∠A = m∠C, and m∠B = m∠D. • The perimeter of parallelogram ABCD is calculated as 2(a + b). •An altitude is drawn from point A to the base DC. •Its area is calculated as a×h.

Trapezoid

A quadrilateral (four-sided polygon) with exactly one pair of parallel sides. •The diagram below shows a typical trapezoid with bases a and b, and height h. •The perimeter of the trapezoid is the sum of the lengths of the sides. •Its area is one-half the product of the height and the sum of the lengths of the bases.

Interval

A range of numeric values; for example, the interval 10-19 indicates all values from 10 through 19, including the endpoints 10 and 19. When grouping statistical data, a data set may be divided into intervals, usually of equal length, spanning the whole range of data values.

Perfect Square

A rational number whose square root is also a rational number. For example 9 = 3 × 3 = 3^2 is a perfect square; so is 0.25 = 0.5 × 0.5 =(0.5)2, and 100 = 10 × 10 = 10^2.

Rational Number

A real number that can be expressed as the ratio of two integers, a/b, with b ≠ 0.

Irrational Number

A real number that cannot be expressed as the ratio of two integers. For example, radical 2 and π are irrational numbers.

Square

A rectangle whose sides are all the same length; a rhombus containing a right angle. •The diagonals of a square are of equal length and are perpendicular bisectors. •The diagram below shows square ABCD with sides of length s. •The perimeter of ABCD is 4s. •Its area is s^2. •The length of a diagonal is d = s^2

Cube

A rectangular solid whose length, width, and depth are equal. •The faces of a cube are squares; each face is the same size. •The volume of a cube whose side is of length s is V = s^3.

Rhombus

A regular parallelogram; that is, a parallelogram whose sides are all of the same length. •In the diagram, rhombus ABCD has sides of length s. •The perimeter of ABCD is 4s. •In general the diagonals of a rhombus are not of the same length except in the special case when the rhombus is a square. •However, the diagonals of a rhombus intersect at right angles and bisect each other; that is, they are perpendicular bisectors, partitioning the rhombus into four congruent right triangles.

Function

A relation in which no two ordered pairs have the same first member. •In a function, each element x of the domain corresponds to exactly one element y in the range. •The notation y = f(x) shows a functional relationship between the independent variable x and the dependent variable y. •The graph of a function passes the vertical line test.

Mapping

A relation pairing each member of the domain with exactly one member of the range.

Construction

A sequence of steps using a compass and a straightedge (an unmarked ruler) to draw construct or draw a geometric object, such as an angle, a line or line segment, a plane figure, etc.

System of Equations

A set of equations with common variables; the solution set of a system of equations satisfies all equations in the system simultaneously. For example, y = 3x + 2 and 2y + 3x = 6 is a system of two linear equations in x and y.

Relation

A set of ordered pairs.

Line

A set of points extending indefinitely in two opposite directions. A line can be uniquely determined by two points. The line containing points A and B is called line AB.

Pythagorean Triple

A set of three positive real numbers, a, b, and c, that satisfy the condition a^2 + b^2 = c^2. These numbers can represent the lengths of the sides of a right triangle with hypotenuse c and legs a and b.

Symbolic Form

A shorthand notation of expressing logical statements. Simple statements are represented by variables, such as a and b. Compound statements combine simple statements using logical connectives.

face

A side of a polyhedron.

Polygon

A simple closed curve formed by line segments. •A three-sided polygon is called a triangle. •A four-sided polygon is called a quadrilateral. •A five-sided polygon is called a pentagon. •A six-sided polygon is called a hexagon. •An eight-sided polygon is called a octagon.

three-dimensional figure

A solid figure such as a polyhedron or a sphere.

Pyramid

A solid whose base is a polygon such as a triangle or a square, and whose lateral surfaces (sides) meet at a point (apex). The diagram shows a pyramid with a square base.

Cone

A solid with a circular base and a lateral surface that comes to an apex point. •The axis of a cone is a line segment joining the center of the base to the apex. •Below, left, is a sketch of a right cone; its axis is perpendicular to its base. •The volume of a right cone with a base of radius r and height h is V= 1/3πr^2h . •Below, right, is a sketch of an oblique cone; its axis is not perpendicular to its base.

SAA

A theorem stating a condition for the congruence of two triangles. ΔA and ΔB are congruent if two angles and the side opposite to one of these angles in ΔA are congruent to the corresponding portions of ΔB.

Cylinder

A solid with two flat, parallel, circular surfaces which form its bases, and a curved, lateral surface. •The axis of a cylinder is a line segment joining the center of one base to the center of the other base. •Below, left, is a sketch of a right cylinder; its axis is perpendicular to its bases. •The volume of a right cylinder with a base of radius r and height h is V = πr2h. •Below, right, is a sketch of an oblique cylinder; its axis is not perpendicular to its bases.

Rectangle

A special case of a parallelogram in which adjacent sides form right angles. •The diagram shows rectangle ABCD with the right angle at D marked. •If the dimensions of the rectangle are length l and width w, then the perimeter of the rectangle is 2l + 2w, and its area is l×w. •The diagonals of a rectangle are congruent.

Reflection in the Line y = x

A special case of reflection in a line. •Reflection in the line y = x is a transformation that maps the point (x, y) onto the point (y, x). •The functional notation for this transformation is ry=x. •The diagram shows the point P and its image P' after reflection in the line y = x.

Reflection in the y-axis

A special case of reflection in the line x = 0. •Reflection in the y-axis is a transformation that maps the point (x, y) onto the point (−x, y ). •The diagram shows the point P and its image P' after reflection in the y-axis. •The functional notation for this transformation is ry-axis.

Reflection in the x-axis

A special case of reflection in the line y = 0. •Reflection in the x-axis is a transformation that maps the point (x, y) onto the point (x, −y ). •The diagram shows point P and its image P' after reflection in the x-axis. •The functional notation for this transformation is r x-axis.

Reflection in the Origin

A special case of reflection in the point (0, 0). •Reflection in the origin is a transformation that maps the point (x, y) onto the point (−x, −y). •In the diagram, P' is the reflection of point P in the origin.

Compound Statement

A statement formed by combining two or more simple statements using logical connectives. For example, "a and b" is a compound statement formed from the statements a and b.

Theorem

A statement that can be proved true. The study of geometry involves many theorems.

Parabola

A symmetrical U-shaped curve. •The graph of the quadratic equation y = ax^2 + bx + c, a ≠ 0, is a parabola that is symmetric about the line: x = -b/2a •This line is also known as the axis of symmetry. •The turning point is the point of intersection of the parabola and its axis of symmetry. •If a > 0, the shape of the parabola is concave up with a minimum at the turning point, as shown below, right. •If a < 0, the parabola is concave down with a maximum at the turning point, as shown below, left. •A parabola can also be defined as the set of all points P in a plane that are equidistant from a fixed line and a fixed point not on the line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Vertical Line Test

A test applied to a graph to determine whether the relation it represents is also a function. Pass a vertical line from left to right across the coordinate plane containing the graph of a relation. If the line never intersects the graph in more than one point, the graph represents a function. The graph of a circle below, left, fails the vertical line test; it does not represent a function. The graph of a parabola below, right, passes the vertical line test; it represents a function.

H.L.

A theorem stating a condition for the congruence of two right triangles. Right triangles ΔA and ΔB are congruent if the hypotenuse and one leg of ΔA are congruent to the corresponding sides of ΔB.

SSS

A theorem stating a condition for the congruence of two triangles. ΔA and ΔB are congruent if the sides of ΔA are congruent to the corresponding sides of ΔB.

ASA

A theorem stating a condition for the congruence of two triangles. ΔA and ΔB are congruent if two angles and the included side in ΔA are congruent to the corresponding portions of ΔB.

AAS

A theorem stating a condition for the congruence of two triangles. ΔA and ΔB are congruent if two angles and the side opposite to one of these angles in ΔA are congruent to the corresponding portions of ΔB.

Rotation

A transformation in which a figure is moved about a fixed point by a given angle measurement. The size of the figure does not change. For example, the diagram shows a transformation in which the black triangle is rotated counterclockwise 90° about the origin. The image of the black triangle under this transformation is shown in blue. The functional notation for a counterclockwise rotation n° about the origin is R n° where n > 0. If n < 0, the rotation is clockwise about the origin.

Dilation

A transformation in which a figure is shrunk or enlarged (scaled) by a constant factor, k ≠ 0. •Under a dilation, the point (x, y) is mapped onto the point (kx, ky). •The transformation can be applied to entire figures. For example, the diagram shows a rectangle dilated by a scale factor of 2. •The functional notation for a dilation by a scale factor k is Dk. •Size is not preserved under dilation unless k = ±1.

Opposite Transformation

A transformation in which the image does not have the same orientation as the original plane figure.

Direct Transformation

A transformation in which the image has the same orientation as the original plane figure.

Isometry

A transformation that preserves distance; that is, the image is the same size as the original.

Direct Isometry

A transformation that preserves orientation (order) and size (isometry).

Opposite Isometry

A transformation that preserves size (isometry), but does not preserve orientation. For example, the orientation may change from clockwise to counterclockwise. •Also called an indirect isometry. •Reflections in a line are opposite isometries.

indirect isometry

A transformation that preserves size (isometry), but does not preserve orientation. For example, the orientation may change from clockwise to counterclockwise. •Also called an opposite isometry. •Reflection in a line is an indirect isometry.

Isosceles Trapezoid

A trapezoid in which the nonparallel sides are of equal length (congruent ). •The diagram shows an isosceles trapezoid with congruent sides line AB and DC, and bases AD and BC. •The diagonals of an isosceles trapezoid are congruent. •The base angles are also congruent.

Acute Triangle

A triangle in which all three angles are acute angles.

Equilateral Triangle

A triangle in which all three sides are congruent, that is, of equal length. •The interior three angles are also congruent; in fact they each measure 60°. •If each side of an equilateral triangle is of length s, the perimeter of the triangle is 3s. •The altitude is h = s square root of 3 / 2.

Scalene Triangle

A triangle in which no two sides have the same length. •In a scalene triangle, no two angles have the same measure. •The diagram shows a scalene triangle.

Circumcenter

A triangle is inscribed in a circle if its vertices are on the circumference of the circle. The circle is circumscribed about the triangle. The circle is called the circumcircle of the triangle, and its center is called the circumcenter of the triangle. This is also the intersection of the three perpendicular bisectors of the three sides of a triangle. In the diagram, point O is the circumcenter of triangle ABC.

Right Triangle

A triangle that contains a right angle. The side opposite the right angle is called the hypotenuse; the other two sides are call the legs. In the right triangle below right, ∠C is the right angle, c is the length of the hypotenuse, a and b are the lengths of the legs. Right triangles have many special properties: •∠A and ∠B are acute and complementary. •The Pythagorean Theorem relates the lengths of the hypotenuse and the legs: c2 = a2 + b2. •Since one leg can be considered the altitude and the other leg the base of the triangle, the area of a right triangle is equal to one-half the product of the lengths of its legs: •The Hypotenuse-Leg Theorem (HL) states that two right triangles are congruent if the hypotenuse and leg of one triangle are congruent to the corresponding parts of the other triangle. •The trigonometric ratios sine, cosine, and tangent relate the lengths of the various sides of a right triangle to the measures of the acute angles

Obtuse Triangle

A triangle that contains an obtuse angle. A given triangle can contain at most one obtuse angle. In the diagram of ΔABC, ∠B is obtuse.

Isosceles Triangle

A triangle with two congruent sides. •The diagram shows an isosceles triangle with congruent sides AB and AC , and base BC . •The congruent base angles, ∠B and ∠C, are shown in red. •The angle opposite the base, ∠A, is called the vertex angle.

Hyperbola

A two-branched curve. •The general equation of a hyperbola with foci at (−c, 0) and (c, 0) is: x^2/a^2 - y^2/b2 image where b^2 = c^2 − a^2. •The graph of a hyperbola consists of two curved branches. •This relation is not a function. The graph of a hyperbola does not pass the vertical line test. A special case is the rectangular hyperbola which has an equation of the form xy = k, for constant k ≠ 0. This equation represents inverse variation between x and y. •The relationship of inverse variation is a function. A graph of a rectangular hyperbola is shown in the diagram to the left.

plane

A two-dimensional surface that extends indefinitely in all directions.

Glide Reflection

A two-step transformation that combines a reflection in a line with a translation along the mirror line

Direct Variation

A type of relationship between two variables: y varies directly as x if y = kx for some constant k ≠ 0. For example, y = 3x. •The graph of direct variation is a straight line.

Translation

A type of transformation in which a point is moved or shifted a given distance in a given direction. •A translation maps the point (x, y) into the point (x + a, y + b). •The size and orientation of a figure do not change under translation, only its position. •The functional notation for this translation is Ta,b. •The diagram shows Δ1 translated to form the image, Δ2.

Venn Diagram

A visual representation of relationships often using circles. For example:

Exterior Angle

An angle whose vertex is one of the vertices of a polygon such that one side of the angle is a side of the polygon, and the other side is an extension of the adjacent side of the polygon. •In the diagram, ∠ACD is an exterior angle of ΔABC. •The exterior angle is supplementary to the adjacent interior angle. •An important theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. In this example, m∠ACD = m∠A + m∠B.

Central Angle

An angle whose vertex is the center of a circle, and whose sides are radii of the circle. For example, in the diagram of the coordinate plane, θ is a central angle of a circle centered at the origin of radius r.

Inscribed Angle

An angle whose vertex lies on a circle and whose sides are chords of the circle. •The measure of an inscribed angle is one-half the measure of the arc that it "cuts" or intercepts. •In the diagram, ∠A is inscribed in circle O. •∠A intercepts arc BC. •m∠A = 1/2 m arc BC.

Minor Arc

An arc of a circle measuring less than 180°. For example, in the diagram of circle O, minor arc is shown in red.

Major Arc

An arc of a circle measuring more than 180°. For example, in the diagram of circle O, major arc is shown in red.

Linear Equation

An equation of degree 1. The general form of a linear equation in two variables is: ax + by = c, where at least one of the constants a, b, and c is nonzero. •The degree of this polynomial is 1. •The graph of this equation is a straight line. •For example, 2x + 1 = 7 is a linear equation in one variable; x + y = 10 is a linear equation in two variables.

Quadratic Equation

An equation of the form ax^2 + bx + c = 0, a ≠ 0. •The graph of a quadratic equation is a parabola. •The roots of this equation are given by the quadratic formula.

Pythagorean Theorem

An equation relating the lengths of the legs and the hypotenuse of a right triangle. In a right triangle with legs of length a and b, and hypotenuse of length c, such as the right triangle shown in the diagram, c^2 = a^2 + b^2.

Proportion

An equation relating two ratios. In the proportion: a/b = c/d where a ≠ 0 and b ≠ 0: •a and d are called the extremes. •b and c are called the means. •The product of the means is equal to the product of the extremes: ad = bc. •This fact forms the basis for solving a proportion, called cross-multiplication

Prime Number

An integer greater than 1 whose only factors are itself and 1. The first few prime numbers are 2, 3, 5, 7, 9, 11, 13.

Even Number

An integer that is evenly divisible by 2, such as −4, −2, 0, 2, 4,... . •If n is any integer, then 2n is always even. •If n is an even integer, then the next consecutive even integer is n + 2.

Odd Number

An integer that is not evenly divisible by 2, such as −3, −1, 1, 3,... . •If n is any integer, then 2n + 1 is always odd. •If n is an odd integer, then the next consecutive odd integer is n + 2

Pi

An irrational number, denoted by π, that is the ratio of the circumference of a circle to its diameter

Ellipse

An oval-shaped curve. •The general equation of an ellipse with foci at (0, c) and (0, −c) is: x^2/a^2 + y^2/b^2 = 1 where b2 = a2 − c2. •This relation is not a function. The graph of an ellipse does not pass the vertical line test.

Congruent Angles

Angles that have equal measures. In the diagram below, "angle A is congruent to angle B" is expressed as ∠A ≅ ∠B.

Conjunction, Law of

If p and q are both given, then the conjunction p q is true.

Reflection in a Point

If point A is reflected in point P, then P is the midpoint of the line segment joining A and its image A'.

Conjunctive Simplification, Law of

If the conjunction "a and b" is true, then the conjuncts a and b are also true. In symbolic form: (a ^ b) → a and (a ^ b) → b

Slope Equation

If two points (x1, y1) and (x2, y2) lie on line L and x1 ≠ x2, the slope, m, of line L is given by the slope equation: m = y2-y1/ x2-x1 •The slope of a horizontal line is 0. •The slope of a vertical line is undefined.

Inverse of a Function

If y = f(x) is a function, then the inverse function is x = f(y), which is formed by interchanging x and y. •The inverse of function f(x) is written as f ^−1(x). •The graph of y = f ^−1(x) is the reflection of the graph of y = f(x) in the line y = x.

Inverse Variation

If y varies inversely as x, then xy = k for some nonzero constant k. •Inverse variation can also be expressed as: y = k/x for x ≠ 0. •For example, the equation xy = 1 is an example of inverse varation. •The graph of an inverse variation relationship is a rectangular hyperbola, as shown in the diagram below. •This relation is a function; it passes the vertical line test.

Antecedent

In a conditional statement "if a then b", a is the antecedent or hypothesis; the part of the statement that follows the "if". For example, in the conditional statement "If it rains, then the picnic is cancelled," the antecedent is "It rains."

Consequent

In a conditional statement "if a then b," b is the consequent or conclusion. It is the part of the statement that follows the "then." For example, in the conditional statement "If it rains, then the picnic is cancelled," the consequent is "The picnic is cancelled."

Simplification, Law of

In a conjunction, either of the conjuncts can be separated out and is true. For example, p ˄ q → p and p ˄ q → q.

Corresponding Sides

In a pair of congruent polygons, corresponding sides "match" and thus have the same length. In a pair of similar polygons, corresponding sides "match" and are proportional in length.

x-Axis

In the coordinate plane, the horizontal axis formed by a real number line. •The x-axis and the y-axis intersect at the origin whose coordinates are (0, 0).

y-Axis

In the coordinate plane, the vertical axis formed by a real number line. •The x-axis and the y-axis intersect at the origin whose coordinates are (0, 0).

Fraction

In the fraction a/b, the ratio of the real numbers a and b with b ≠ 0, a is the numerator and b is the denominator. An algebraic fraction is the ratio of two algebraic expressions.

Abscissa

In the ordered pair (a, b), the first member of the pair, a.

ordinate

In the ordered pair (a, b), the second member of the pair, b.

Cross Multiplication

In the proportion: a/b = c/d , a and d are called the extremes, whereas b and c are called the means. Cross-multiplication makes use of the fact that the product of the means is equal to the product of the extremes: ad = bc. It is a useful technique for solving equations involving a proportion.

Lateral Area

In three-dimensional figures, the surface area of the sides of the figure, not including the base(s).

Real Number

Integers, rational numbers and irrational numbers are all real numbers. •There is a one-to-one correspondence between the set of real numbers and the points on a real number line. •Every point on the real number line corresponds to exactly one real number; and every real number corresponds to exactly one point on the real number line.

Alternate Interior Angles

Interior angles lying on opposite sides of a transversal. Pairs of alternate interior angles have the same measure. In the diagram below, transversal crosses parallel lines and , intersecting them at points G and H, respectively. ∠AGF and ∠EHD form a pair of alternate interior angles, and ∠AGF ≅ ∠EHD.

Congruent Segments

Line segments that have the same length. "Line segment AB is congruent to line segment CD" is written as line AB ≅ line CD.

Oblique

Lines or surfaces that are neither perpendicular nor parallel to each other; a solid in which the axis is not perpendicular to the plane of the base.

Parallel Lines

Lines that lie in the same plane, but do not have any points in common; they do not intersect. In the diagram, "line AB is parallel to line CD" is written as: line AB ll line CD

Truth Value

Logical statements can have one of two possible truth values: true, usually denoted T, and false, usually denoted by F.

Like Terms

Monomial terms that include powers of the same base but have different coefficients; for example, 5x^2y and -8x^2y are like terms. Like terms can be added or subtracted by adding or subtracting their coefficients. For example 5x^2y 8x^2y = - 3x^2y. The process of solving an equation often involves collecting and combining like terms in this way.

Perpendicular Planes

Planes in which any two intersecting lines, one in each plane, form a right angle. In the diagram below, plane P is perpendicular to plane Q.

Non-collinear points

Points that do not lie on the same line.

Collinear Points

Points that lie on the same line. In the diagram below, points A, B, and C are collinear, which is expressed as: line ABC

lateral

Refers to a side, such as the side of a prism.

Reflection in a Line

Reflection in line l is a transformation that maps point P lying on one side of l at distance d onto point P′ lying on the opposite side of l, also at distance d, in such a way that line l forms the perpendicular bisector of the line segment joining P and its image P′. Put simply, if you "fold" the diagram along line l, point P and its image P′ are superimposed. The transformation maps a point lying on l onto itself.

Law of Cosines

Relates the measures of the sides of a triangle to the measures of its interior angles. The Law of Cosines expresses the length of one side of a triangle in terms of the lengths of the other two sides and the cosine ratio of the opposite angle:

Factoring

Rewriting an algebraic expression as the product of one or more factors. •For example, the trinomial x^2 + x - 2 can be factored into the product of the binomials (x + 2) and (x - 1). •The polynomial x^3 + 4x^2 + x - 6 can be factored into the product of (x^2 + x - 2) and (x + 3).

Angle of Depression

The angle between the line of sight and an object below the line of sight. For example, in the diagram, represents a surface, represents an object below the surface, and is the line of sight. Angle θ is the angle of depression, that is, the angle that the line of sight from point A to the bottom of the object makes with the surface.

Angle of Elevation

The angle between the line of sight and an object that is above the line of sight. For example, in the diagram, represents a tree, represents the horizon, and is the line of sight. Angle θ is the angle of elevation, that is, the angle that the line of sight from point A to the top of the tree makes with level ground.

Included Angle

The angle between two sides of a triangle. In the diagram, angle B, marked in red, is included between sides line AB and line BC, marked in blue.

Vertex Angle

The angle opposite the base of an isosceles triangle. •The angles opposite the two congruent sides of an isosceles triangle are called the base angles. •The diagram shows an isosceles triangle with congruent sides AB and AC, and base BC. •The vertex angle, ∠A, is shown in red. ∠B and ∠C are the base angles.

Area of Triangle

The area of a triangle with base of length b and height h is: A = 1/2 bh.

Equivalent Equations

The basic properties of equality for real numbers a, b, and c are: if a = b, then a + c = b + c and ac = bc. The application of these properties enables us to solve equations by collecting all terms with variables on one side of the equation, and all constants on the other side of the equation, to form a new equation equivalent to the original equation. For example, given the equation 3x − 4 = 5 we can add the constant 4, which is the additive inverse of -4, to each side of the equation: 3x − 4 + 4 = 5 + 4 to form the equivalent equation: 3x = 9. We can then multiply each side of the equation by 1/3 , the multiplicative inverse of 3: (1/3) 3x = (1/3) 9 and apply the associative property: (1/3 X 3) x = 3 to form the equivalent equation: x = 3, which is "solved" for the variable x.

Incenter

The center of an incircle. This is also the intersection of the three angle bisectors of a triangle. In the diagram, the incenter of the triangle is shown in red.

Contrapositive

The compound statement formed by negating and interchanging the antecedent and the consequent of a conditional statement. The contrapositive of the conditional statement "if a then b" is "if not b then not a." For example, consider the conditional statement, "If it is sunny, then we will go on a picnic." Its contrapositive is "If we do not go on a picnic, then it is not sunny." If a conditional statement is true, then so is its contrapositive; likewise, if a conditional statement is false, its contrapositive is also false. In symbolic form, the contrapositive of the conditional statement a → b is ~b → ~a. A conditional statement and its contrapositive always have the same truth value, as shown in the truth table below.

Quadrant

The coordinate plane is divided into four quarters, or quadrants, labeled in the diagram below as I, II, III, and IV. •If the point (x, y) lies in Quadrant I, then x > 0 and y > 0. •If (x, y) lies in Quadrant II, then x < 0 and y > 0. •If (x, y ) lies in Quadrant III, then x < 0 and y < 0. •If (x, y) lies in Quadrant IV, then x > 0 and y < 0.

Measure of an Angle or Arc

The degree (°) is a unit of measure of an angle or an arc. •A straight angle is measures 180°. •A complete circle is an arc measuring 360°. •The degree measure of angle A is written as: m∠A. •The degree measure of arc BC is written as: m ̑͡BC

Measure of an Angle

The degree (°) is a unit of measure of an angle. •A straight angle measures 180°. •A protractor is an instrument that can be used to measure an angle. •The degree measure of angle A is written as: m∠A.

Arc Length

The distance between the endpoints of an arc, measured in linear units.

Slant Height

The distance from a vertex to the base of a pyramid or cone, measuring along the side (lateral surface). The cone in the diagram has a height (altitude) of 8 feet, but a slant height of 10 feet.

Perimeter

The distance measured around a plane figure. •The perimeter of a polygon is the sum of the lengths of its sides. •The perimeter of a circle is called its circumference.

Vertex

The endpoint common to the two rays forming an angle. In the diagram, point A is a vertex. An angle is often referred to simply by its vertex, for example, ∠A.

Point-slope Form

The equation of a line in the form y - y1 = m(x - x1) where m is the slope and (x1, y1) is a point on the line.

Axis of Symmetry

The graph of a parabola is symmetric about a line called axis of symmetry. •For the parabola whose equation is y = ax2 + bx + c, the axis of symmetry is a vertical line with the equation: x= -b/2a •The axis of symmetry and the parabola intersect at the turning point. •The parabola in the diagram below is symmetric about the y-axis.

Turning Point of a Parabola

The point of intersection of a parabola and its axis of symmetry. •If the shape of the parabola is concave down, as in the diagram below, left, the parabola reaches its maximum value at the turning point. •If the shape of the parabola is concave up, as in the diagram below, right, the parabola reaches it minimum value at the turning point.

Centroid

The point where the three medians of a triangle intersect. In the diagram of ΔTEM, the medians line AB , line BT and CE intersect at point D, the centroid.

x-intercept

The point(s) on a graph where it intersects the x-axis. On the graph of the parabola, the x-intercepts are marked in red.

y-Intercept

The point(s) on a graph where it intersects the y-axis. •For a linear equation written in slope-intercept form, y = mx + b, the slope of the line is m, and the y-intercept is b. •For example, in the graph to the left, line l intersects the y-axis at the point (0, −4). That means the y-intercept of line l is −4. •The y-intercept for a vertical line is undefined. •The graph of a parabola may also have a y-intercept. For the equation of a parabola of the form y = ax^2 + bx + c, a ≠ 0, the y-intercept is c.

endpoints

The points at the beginning and end of a line segment. In the diagram of line AB , the endpoints, A and B, are marked in red.

Solving an Equation

The process of finding the solution set of an equation, that is, the value or values that a variable may have which makes an equation a true statement. For example, for the equation 2x − 4 = 0, the value x = 2 makes the equation a true statement, so 2 is a root of the equation. The roots of the equation x^2 = 9 are 3 and −3. •Methods for solving equations apply the associative, distributive, and other properties of real numbers to form equivalent equations. •A general strategy is to combine and collect terms involving the variable on the left hand side of the equation, and constants on the right side.

Rate

The ratio of the change in one variable or quantity to another. For example, the speed of an object is the ratio of the change in its distance to the time it takes to travel that distance, expressed as s = d/t.

Coordinate

The real number corresponding to a point on a real number line. For example, the coordinate of the red point on the number line below is 2. In the rectangular coordinate system, every point on the plane corresponds to a coordinate pair (a, b) which describes the location of the point with respect to a horizontal and a vertical axis. For example, the coordinates of the red point in the coordinate plane below are (3, 2), indicating that the point lies +3 along the horizontal number line and +2 along the vertical number line.

Coordinate(s)

The real number corresponding to a point on a real number line. For example, the coordinate of the red point on the number line below is 2. In the coordinate plane, every point on the plane corresponds to an ordered pair of real numbers (a, b) that describes the location of the point with respect to a horizontal and a vertical axis. For example, the coordinates of the red point in the coordinate plane to the left are (3, 2). The point lies +3 along the horizontal number line and +2 along the vertical number line.

Domain

The replacement set; the set of possible replacement values for a variable. For example, the domain might be the set of integers or the set of real numbers. In a relation, the domain is the set of all first members of the ordered pairs comprising the relation.

Quadratic Formula

The roots of the quadratic equation ax^2 + bx + c = 0, a ≠ 0, are x=-b+radical b2-4ac/2a. The radicand b^2 − 4ac is called the discriminant. The nature of the discriminant tells us about the roots of the quadratic equation: •If b^2 − 4ac > 0, then the quadratic has two, real roots. If, in addition, b2 − 4ac is a perfect square, the roots are rational; otherwise they are irrational. •If b^2 − 4ac = 0, then the quadratic has one, real (double) root. •If b^2 − 4ac < 0, then the roots of the quadratic are imaginary (not real).

Logic

The science of reasoning.

Graph

The set of all points on a number line or in the coordinate plane whose coordinates satisfy a set of conditions, such as an equation, a system of equations, an inequality, or a system of inequalities.

Solution Set

The set of all possible values of a variable that make a statement about the variable true. For example, the solution set of the quadratic equation x^2 = 9 is {−3, 3}.

Locus of Points

The set of points that satisfy a given set of conditions. The plural of locus is loci.

Compound Locus

The set of points that satisfy two or more conditions; the points common to two or more loci (the plural of locus).

Included Side

The side between two angles of a triangle. In the diagram, side AC , marked in red, is included between angle A and C, marked in blue.

base (geometric)

The side of a figure that is perpendicular to the altitude.

Hypotenuse

The side opposite the right angle of a right triangle. In the right triangle below, the hypotenuse is shown in red.

Radical

The square root of a nonnegative real number a is a real number b such that b^2 = a. The square root of a is written as radical a where radical is called the square root or radical sign. Here are some rules for working with radical expressions: •(radical a^2) =radical a^2 =a • radical ab = radical a radical b • radical a/b = radical a/b

Square Root

The square root of a real number a > 0 is a real number b such that b^2 = a. is called the square root or radical sign. For example, = ±5, because 5×5 = 25 = (-5)×(-5). Sometimes, only the positive, or principal square root is desired. There are several properties which can be used to simplify radical expressions:

Double Negation, Law of

The statements a and ~(~a) are logically equivalent, expressed symbolically as: a ↔ (~a).

Surface Area

The total area of all of the surfaces of a polyhedron or other 3-dimensional object combined, including sides (lateral area) and bases.

Base Angles

The two angles opposite the two congruent sides of an isosceles triangle. The diagram shows an isosceles triangle with congruent sides line AB and line AC, and Base line BC. The congruent base angles, ∠B and ∠C, are marked in red. The angle opposite the base, ∠A, is called the vertex angle.

Conjunct

The two parts of a conjunction. For example, in the conjunction "a and b," written symbolically as a b, statements a and b are the conjuncts.