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

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

G–L

geometric mean

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

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

The longest side in a right triangle.

The side opposite the right angle.

image

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

incenter

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 2 Lesson 2, Unit 3 Lesson 10

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

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

joint events

Events that can occur at the same time.

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

kite

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

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

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

line of symmetry

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

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

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

line segment

A piece of a line with two endpoints.

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

A line segment has length and can be measured.

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

linear pair

Unit 1 Lesson 5, Unit 1 Lesson 6, Unit 4 Lesson 6

Two supplementary angles that share a vertex and a side.

A linear pair always make a line.

M–R

magnitude of a vector

Unit 7 Lesson 12

The length of a vector.

See vector.

major axis, minor axis of an ellipse

Unit 7 Lesson 10

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

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

median in a triangle

Unit 3 Lesson 4

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

midline of a triangle

Unit 4 Lesson 2

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

midline of a triangle theorem

Unit 4 Lesson 2

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

midpoint

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

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

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

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

See also bisect.

model, mathematical

Unit 4 Lesson 11

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

mutually exclusive

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

mutually exclusive event

Unit 1 Lesson 6, Unit 6 Lesson 1

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

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

n-gon

A polygon with $n$ number of sides.

See polygon.

obtuse angle / obtuse triangle

Unit 1 Lesson 2, Unit 8 Lesson 6

octagon

An eight-sided polygon.

opposite (or negative) reciprocal slope

Unit 1 Lesson 5, Unit 1 Lesson 6

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

opposite angles, opposite vertices

Opposite angles in a quadrilateral do not share a side.

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

opposite side in a triangle

Unit 1 Lesson 5, Unit 1 Lesson 6

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

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

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

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

orientation

Unit 1 Lesson 1, Unit 1 Lesson 3

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

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

origin

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

parabola: conic definition, geometric definition

Unit 7 Lesson 7

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

parallel line

parallelogram

A quadrilateral in which the opposite sides are parallel.

pentagon

Unit 1 Lesson 4, Unit 3 Lesson 3, Unit 3 Lesson 4

A five-sided polygon.

perpendicular bisector

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

perpendicular lines

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

plane

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

point

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

point of concurrency

Unit 3 Lesson 10

See concurrent lines.

polygon

Unit 3 Lesson 1, Unit 3 Lesson 5

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

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

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

postulate

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

pre-image / image

Unit 6 Lesson 6, Unit 6 Lesson 8

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

preserves distance and angle measure

Unit 1 Lesson 4

Measurements are not changed under a rigid transformation.

prism: right, oblique

Prism: Also called a polyhedron.

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

proof by contradiction

Unit 2 Lesson 4

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

proof: types—flow, two-column, paragraph

properties of equality

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

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

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

proportion: proportionality statement

Unit 4 Lesson 4

A proportion is a statement that two ratios are equal.

pyramid

Unit 6 Lesson 7

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

Pythagorean theorem

Unit 1 Lesson 4, Unit 1 Lesson 5, Unit 1 Lesson 7

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

quadrilaterals: types

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

quantity

Unit 8 Lesson 4

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

radian

Unit 6 Lesson 4

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

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

radii

Unit 2 Lesson 1

Plural of radius. See circle.

ratio

Unit 4 Lesson 4, Unit 4 Lesson 6

A ratio compares the size or amount of two values.

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

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

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

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

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

ray

Unit 2 Lesson 1

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

Notation: $\vec{A B}$ – ray $A B$

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

reasoning – deductive/inductive

Unit 3 Lesson 1

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

Two Types of Reasoning

Inductive reasoning:

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

Deductive reasoning:

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

Observations

General Premise

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

Conclusion:

Opposite angles formed by intersecting lines are always congruent.

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

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

rectangle

See quadrilaterals: types.

reference angle

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

reflection

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

The orientation of the image is reversed.

regular polygon

See polygon.

rhombus

A quadrilateral in which all sides are congruent.

right angle

An angle that measures $90 °$ .

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

rigid transformation

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

rotation

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

rotational symmetry

See symmetry.

S–X

same-side interior angles

See angles made by a transversal.

SAS triangle similarity

Unit 4 Lesson 1, Unit 6 Lesson 6

See triangle similarity.

scalar quantity

Unit 7 Lesson 12

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

scale factor

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

scalene triangle

Unit 3 Lesson 10, Unit 5 Lesson 1

A triangle that has three unequal sides.

secant line (in a circle), tangent line

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

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

sector

Unit 6 Lesson 3

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

A pie-shaped part of a circle.

segment of a circle

Unit 6 Lesson 3

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

side-splitter theorem

Unit 4 Lesson 4

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

similarity

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

solids of revolution

Unit 8 Lesson 2

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

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

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

special right triangles

Unit 8 Lesson 5

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

square

See quadrilaterals: types.

SSS triangle similarity

See triangle similarity.

straight angle

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

summation notation

Unit 6 Lesson 9

supplementary angles

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

symmetric

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

symmetry

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

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

tessellation

Unit 3 Lesson 1, Unit 3 Lesson 5

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

theorem

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

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

translation

A translation is a rigid transformation.

transversal

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

trapezoid

A quadrilateral with exactly one pair of parallel opposite sides.

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

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

tree diagram

Unit 9 Lesson 1

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

triangle congruence criteria: ASA, SAS, AAS, SSS

Unit 2 Lesson 4

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

ASA stands for “angle-side-angle.”

SAS stands for “side-angle-side.”

AAS stands for “angle-angle-side.”

SSS stands for “side-side-side.”

triangle similarity

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

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

AA Similarity

SAS Similarity

SSS Similarity

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

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

$sine A$ abbreviated $\sin A$

$cosine A$ abbreviated $\cos A$

$tangent A$ abbreviated $\tan A$

A trigonometric ratio always includes a reference angle.

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

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

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

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

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

two-column proof

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

two-way table

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

union

Unit 9 Lesson 3

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

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

vector, vector quantity

Unit 7 Lesson 12

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

Notation: $\vec{A B}$

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

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

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

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

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

vertical angles