Geometry – Glossary | Open Up HS Math (CCSS), Student

Unit 3 Lesson 4, Unit 4 Lesson 7, Unit 6 Lesson 6

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

See also angles made by a transversal.

altitude

Altitude of a triangle:

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

Altitude of a solid:

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

Ambiguous Case of the Law of Sines

Unit 8 Lesson 8

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

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

angle

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

angle bisector

Unit 3 Lesson 4

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

angle of depression/angle of elevation

Unit 4 Lesson 10

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

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

angle of rotation

Unit 5 Lesson 1, Unit 5 Lesson 4

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

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

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

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

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

angles made by a transversal

Unit 5 Lesson 5, Unit 6 Lesson 3

arc length

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

Equation for finding arc length:

$s = r θ$

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

arc of a circle, intercepted arc

Unit 5 Lesson 1, Unit 5 Lesson 3

Arc: A portion of a circle.

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

asymptote

Unit 7 Lesson 11

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

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

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

auxiliary line

Unit 2 Lesson 5

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

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

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

Unit 1 Lesson 6

To divide into two congruent parts.

A bisector can be a point or a line segment.

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

Cavalieri's principle

Unit 6 Lesson 8

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

center of dilation

Unit 4 Lesson 1

See dilation.

central angle

Unit 5 Lesson 1

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

centroid

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

chord of a circle

Unit 5 Lesson 1

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

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

circle

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

Notation: $⨀ D$

circle: equation in standard form; equation in general form

Unit 7 Lesson 5

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

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

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

circumcenter

Unit 2 Lesson 2, Unit 3 Lesson 10

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

circumscribe

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

All of these polygons are inscribed in the circles.

clockwise / counterclockwise

Unit 1 Lesson 1

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

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

coincides (superimposed or carried onto)

Unit 1 Lesson 3, Unit 2 Lesson 4

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

collinear, collinearity

Unit 4 Lesson 1

When three or more points lie in a line.

Note: Any two points can define a line.

Noncollinear: Not collinear.

complement (in probability)

Unit 9 Lesson 3

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

complementary angles

Unit 4 Lesson 9

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

concave and convex

Unit 4 Lesson 5

Polygons are either convex or concave.

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

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

concentric circles

Circles with a common center.

concurrent lines

Unit 6 Lesson 6, Unit 6 Lesson 8

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

Point $M$ is the point of concurrency.

conditional probability

Unit 9 Lesson 1

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

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

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

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

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

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

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

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

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

conditional statement

Unit 3 Lesson 4

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

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

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

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

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

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

cone: right, oblique

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

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

congruence statement

Unit 2 Lesson 1

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

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

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

congruent (CPCTC)

Unit 1 Lesson 1, Unit 1 Lesson 5

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

The symbol for congruent is $≅$ .

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

conjecture

Unit 1 Lesson 7

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

construction

Unit 2 Lesson 1

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

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

Constructing an angle bisector:

convergence

Unit 6 Lesson 2

Moving toward or approaching a definite value or point.

converse statement

Unit 3 Lesson 4, Unit 3 Lesson 7

See conditional statement.

corresponding angles

Unit 2 Lesson 3, Unit 3 Lesson 6

Angles that are in the same relative position.

corresponding parts (in a triangle)

Unit 2 Lesson 4

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

corresponding points / sides

Unit 1 Lesson 3, Unit 2 Lesson 3

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

counterexample

Unit 2 Lesson 4

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

Statement: All blondes drive red cars.

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

CPCTC

Unit 2 Lesson 5

See congruent (CPCTC).

cross-section of a solid

Unit 8 Lesson 1

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

cyclic polygon

Unit 5 Lesson 3

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

cylinder: right, oblique

Unit 6 Lesson 6, Unit 6 Lesson 8

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

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

definition

Unit 3 Lesson 5

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

degree

Unit 1 Lesson 5, Unit 4 Lesson 5

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

density

Unit 8 Lesson 4

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

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

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

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

diagonal

Any line segment that connects nonconsecutive vertices of a polygon.

dilation

Unit 4 Lesson 1

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

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

directed distance

Unit 4 Lesson 6, Unit 7 Lesson 12

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

direction of a vector

Unit 7 Lesson 12

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

See vector.

directrix

Unit 7 Lesson 7

See parabola.

disc or disk

Unit 8 Lesson 2

See solid of revolution.

disjoint

Unit 9 Lesson 4

See mutually exclusive.

edge / face / vertex of a 3-D solid

Unit 8 Lesson 1

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

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

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

ellipse

Unit 7 Lesson 10

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

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

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

equality statements

Unit 2 Lesson 1

A mathematical sentence that states two values are equal.

It contains an equal sign.

equidistant

Unit 3 Lesson 5

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

equilateral, equilateral triangle

Unit 1 Lesson 2, Unit 1 Lesson 5

Equilateral means equal side lengths.

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

exterior angle of a triangle (remote interior angles)

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

exterior angle theorem

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

false negative/positive

Unit 9 Lesson 1

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

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

flow proof

Unit 3 Lesson 4

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

focus

Unit 7 Lesson 7

See parabola.

frustum

Unit 8 Lesson 3

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

G–L

geometric mean

Unit 4 Lesson 7

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

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

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

geometric series

Unit 6 Lesson 9

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

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

hexagon

Unit 1 Lesson 5

A six-sided polygon.

hyperbola

Unit 7 Lesson 11

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

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

hypotenuse

Unit 4 Lesson 8

The longest side in a right triangle.

The side opposite the right angle.

image

Unit 1 Lesson 1

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

incenter

Unit 2 Lesson 2, Unit 3 Lesson 10

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

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

independent event / dependent event

Unit 9 Lesson 4

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

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

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

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

inscribed angle

Unit 5 Lesson 1

See angles associated with circles.

inscribed in a circle

intersection of sets

Unit 9 Lesson 3

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

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

inverse trigonometric ratio

Unit 4 Lesson 10

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

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

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

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

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

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

isosceles triangle, trapezoid

Unit 1 Lesson 2

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

joint events

Unit 9 Lesson 4

Events that can occur at the same time.

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

kite

Unit 5 Lesson 4

A quadrilateral with two pairs of congruent, adjacent sides.

law of cosines

Unit 8 Lesson 7

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

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

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

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

The law of cosines is useful for finding:

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

law of sines

Unit 8 Lesson 6

limit (convergence)

Unit 6 Lesson 2

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

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

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

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

line

Unit 1 Lesson 2

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

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

line of symmetry

Unit 1 Lesson 5

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

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

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

line segment

Unit 1 Lesson 2

A piece of a line with two endpoints.

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

A line segment has length and can be measured.

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

linear pair