Alternate interior angles are created when 2 parallel lines are crossed by another line. This line is called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

This diagram shows 2 pairs of alternate interior angles: 

• Angles \(a\) and \(d\)
• Angles \(b\) and \(c\)

There are two horizontal parallel lines, and a third diagonal line drawn from the bottom left to the upper right, intersecting both horizontal lines. The diagonal line is labeled transversal. There are four angles created by the diagonal line inside the parallel lines. The upper left angle is labeled a, upper right is b, lower left is c, and lower right is d.

A base is a specific face of a prism or pyramid.

A prism has 2 identical bases that are parallel. A pyramid has 1 base.

A prism or pyramid is named for the shape of its base.

In expressions like \(5^3\) and \(8^2\), the 5 and the 8 are bases. They tell what factor is multiplied repeatedly. For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).

A coefficient is a number that is multiplied by a variable.

• In the expression \(3x+5\), the coefficient of \(x\) is 3.
• In the expression \(y+5\), the coefficient of \(y\) is 1, because \(y=1 \boldcdot y\).

Complementary angles have measures that add up to 90\(^\circ\).

For example, a \(15^\circ\) angle and a \(75^\circ\) angle are complementary.

A cone is a three-dimensional figure like a pyramid, but the 1 base is a circle.

One figure is congruent to another if it can be moved with translations, rotations, and reflections to fit exactly over the other.

In this figure, Triangle A is congruent to Triangles B, C, and D.

• A translation takes Triangle A to Triangle B.
• A rotation takes Triangle B to Triangle C.
• A reflection takes Triangle C to Triangle D.

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

In this example, the constant of proportionality is 3.

In an expression like \(5x+2\), the number 2 is called the constant term. It doesn’t change when the variable \(x\) changes.

• In the expression \(7x+9\), the constant term is 9.
• In the expression \(5x+(\text-8)\), the constant term is -8.
• In the expression \(12-4x\), the constant term is 12.

Corresponding parts are the parts that match up between a figure and its scaled copy. They have the same relative position. Points, segments, angles, or distances can be corresponding.

Point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).

A cross-section is the new face that is seen when a three-dimensional figure is sliced.

A rectangular pyramid is sliced parallel to the base. The cross-section is a smaller rectangle.

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The cube root of a number \(n\) is the number whose cube is \(n\). It is also the edge length of a cube with a volume of \(n\). The cube root of \(n\) is written as \(\sqrt[3]{n}\).

The cube root of 64 is written as \(\sqrt[3]{64}\). Its value is 4 because \(4^3\) is 64. 

\(\sqrt[3]{64}\) is also the edge length of a cube that has a volume of 64. 

A cylinder is a three-dimensional figure like a prism, but with 2 bases that are circles.

A dependent variable represents the output of a function.

For example, someone needs to buy 20 pieces of fruit and decides to buy apples and bananas. If they select the number of apples first, the equation \(b=20-a\) shows the number of bananas they can buy. The number of bananas is the dependent variable because it depends on the number of apples.

A dilation is a transformation that can reduce or enlarge a figure. Each point on the figure moves along a line closer to or farther from a fixed point. That fixed point is the center of the dilation. All of the original distances are multiplied by the same scale factor.

Triangle \(DEF\) is a dilation of triangle \(ABC\). The center of dilation is \(O\). The scale factor is 3.

Every point of triangle \(DEF\) is 3 times as far from \(O\) as every corresponding point of triangle \(ABC\).

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, \(3x+4x\) is equivalent to \(5x+2x\).

• When \(x\) is 3, both expressions equal 21.
• When \(x\) is 10, both expressions equal 70.
• When \(x\) is any other number, both expressions still have equal value.

To expand an expression, use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression.

For example, the expression \(5(4x+7)\) can be expanded to get the equivalent expression \(20x + 35\).

In expressions like \(5^3\) and \(8^2\), the numbers 3 and the 2 are called exponents. They tell how many times a number is used as a factor.

For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).

To factor an expression, use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression.

For example, the expression \(20x + 35\) can be factored to get the equivalent expression \(5(4x+7)\).

A function is a rule that has exactly 1 output for each possible input.

An image is the result of translations, rotations, and reflections on an object. Every part of the original object moves in the same way to match up with a part of the image.

Triangle \(ABC\) has been translated up and to the right to make triangle \(DEF\). Triangle \(DEF\) is the image of the original triangle \(ABC\).

An independent variable represents the input of a function.

For example, suppose someone needs to buy 20 pieces of fruit and decides to buy some apples and bananas. If they select the number of apples first, the equation \(b=20-a\) shows the number of bananas they can buy. The number of apples is the independent variable because any number can be chosen for it.

An inequality is a statement that compares two values or expressions using symbols such as “\(>\)” or “\(<\)”. 

• The symbol “\(>\)” means “is greater than.”
• The symbol “\(<\)” means “is less than.”

For example, the inequality \(\text-3 > \text-7\) tells us that -3 is greater than -7.

An integer is a type of number. All whole numbers and their opposites are integers. 

The labels on this number line show all the integers from -10 to 10.

An irrational number is a number that is not rational. It cannot be written as a positive fraction, a negative fraction, or zero.

Pi (\(\pi\)) and \(\sqrt2\) are examples of irrational numbers.

The legs of a right triangle are the sides that make the right angle. 

Here are some right triangles. Each leg is labeled.

Two quantities have a linear relationship when:

• One quantity changes by a set amount, whenever the other quantity changes by a set amount.
• The rate of change is constant.
• The graph of the relationship is a line.

This graph shows a linear relationship between number of days and number of pages read.

When the number of days increases by 2, the number of pages read always increases by 60. The rate of change is constant, 30 pages per day.

A negative association is a relationship between 2 quantities where one tends to decrease as the other increases. In a scatter plot, the data points tend to group around a line with negative slope.

This scatter plot shows a negative association between the the price of a book and the number of books sold.

An outlier is a data value that is far from the other values in the data set.

This scatter plot shows 1 outlier.

A scatterplot. Horizontal, from 20 to 32, by 2's, labeled foot length in centimeters. Vertical, from 7 to 12, by 1’s, labeled foot width in centimeters. 20 dots trend upward and to the right. Line drawn, trends linearly upward and right with 11 dots above lie and 9 below. No dots lie on the line. The line begins at about point 21 point 9 comma 9 and ends at about 31 point 25 comma 11 point 5.

A positive association is a relationship between 2 quantities where one tends to increase as the other increases. In a scatter plot, the data points tend to group around a line with positive slope.

This scatterplot shows a positive association between dog height and dog weight.

A prism is a type of polyhedron with 2 bases that are identical and parallel. The bases are connected by parallelograms.

Here are some drawings of prisms.

A pyramid is a type of polyhedron that has 1 base. All the other faces are triangles that meet at a single vertex.

Here are some drawings of pyramids.

The rate of change is the amount \(y\) changes when \(x\) increases by 1. On a graph, the rate of change is the slope of the line.

In this graph, \(y\) increases by 15 dollars when \(x\) increases by 1 hour. The rate of change is 15 dollars per hour.

Graph, horizontal axis, x, amount earned in hours. vertical axis, y, amount earned in dollars. line starting at 0 comma 10, passing through 2 comma 40 and 60 comma 100.

A rational number is a number that can be written as a positive fraction, a negative fraction, or zero. It can be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\) is not equal to 0.

For example, 0.7 is a rational number because it can be written as \(\frac{7}{10}\).

Some examples of rational numbers: \(\frac74,0,\frac63,0.2,\text-\frac13,\text-5,\sqrt9\)

Two numbers that multiply to equal 1 are reciprocals.

• 8 and \(\frac{1}{8}\) are reciprocals because \(8 \boldcdot \frac{1}{8} = 1\).
• \(\frac{2}{5}\) is a reciprocal of \(\frac{5}{2}\) because \(\frac{2}{5} \boldcdot \frac{5}{2} = 1\).

A reflection is a transformation that “flips” a figure over a line. Every point on the figure moves to a point directly on the opposite side of the line. The new points are the same distance from the line as they are in the original figure.

This diagram shows a reflection of A over line \(\ell\) that makes the mirror image B.

The relative frequency of a category tells the proportion at which the category occurs in the data set. It is written as a fraction, decimal, or percentage of the total number.

For example, there were 21 dogs in a park. This table shows the frequency and the relative frequency of each color. 

A repeating decimal has digits that keep going in the same pattern over and over. The repeating digits are marked with a line above them.

• The decimal representation for \(\frac13\) is \(0.\overline{3}\), which means 0.3333333 . . . .
• The decimal representation for \(\frac{25}{22}\) is \(1.1\overline{36}\), which means 1.136363636 . . . .

A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations. So is any sequence of these.

A rotation is a transformation that “turns” a figure. Every point on the figure moves around a center by a given angle in a specific direction.

This diagram shows Triangle A rotated around center \(O\) by 55 degrees clockwise to get Triangle B.

A scale tells how the measurements in a scale drawing represent the actual measurements.

The scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches represent 16 feet, and \(\frac12\) inch represents 4 feet.

A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.

To create a scaled copy of a figure, all the side lengths in the original figure are multiplied by the same number. This number is called the scale factor.

In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).

A scaled copy is a copy of a figure where every side length in the original figure is multiplied by the same number.

Triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) is multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).

A scatter plot is a graph that shows values of 2 variables on a coordinate plane. It can be used to look for relationships between the 2 variables.

Each point on this scatter plot represents the height and weight of 1 dog.

Scientific notation is a way to write very large or very small numbers. They are written as a product. The first factor is a number greater than or equal to 1, but less than 10. The second factor is a power of 10.

• The number 425,000,000 in scientific notation is \(4.25 \times 10^8\).
• The number 0.0000000000783 in scientific notation is \(7.83 \times 10^{\text-11}\).

A segmented bar graph shows categories within a data set. Each whole bar represents all the data in one main category. Each bar is separated into parts (segments) that show subcategories.

Stacked bar graph in blue and yellow stripes. Horizontal labeled 10 to 12 years old, 13 to 15 years old and 16 to 18 years old. Vertical labeled 0 to 100, by 25's. Blue section means has cell phone. Yellow stripes means has no cell phone.

This segmented bar graph shows the percentage of people in different age groups that do and do not have a cell phone. For example, among people ages 10 to 12, about 40% have a cell phone and 60% do not.

A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.

This diagram shows a sequence of transformations to move Figure A to Figure C. 

First, A is translated to the right to make B. Next, B is reflected across line \(\ell\) to make C.

Two figures are similar if one can fit exactly over the other after transformations.

This figure shows triangle \(ABC\) is similar to triangle \(DEF\).

• Rotate triangle \(ABC\) around point \(B\). 
• Then dilate it with center point \(O\).
• The image will fit exactly over triangle \(DEF\).

A sequence of transformations. Triangle ABC is rotated around point B and and then dilated with center point O to fit exactly over Triangle DEF. Triangle ABC is similar to triangle DEF.

Slope is a number that describes how steep a line is. To find the slope, divide the vertical change by the horizontal change for any 2 points on the line.

The slope of this line is 2 divided by 3, or \(\frac23\).

A solution to an equation is a number that can be used in place of the variable to make the equation true.

• The solution to the equation \(m+1=8\) is 7, because it is true that \(7+1=8\).
• The solution to \(m+1=8\) is not 9, because \(9+1\) does not equal 8. 

A solution to an equation with 2 variables is a pair of values for the variables that make the equation true.

For example, one solution to the equation \(4x+3y=24\) is \((6,0)\). Substituting 6 for \(x\) and 0 for \(y\) makes this equation true because \(4(6)+3(0)=24\).

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

• A solution to the inequality \(c<10\) is 5, because it is true that \(5<10\).
• Some other solutions to this inequality are 9.9, 0, and -4.

A sphere is a three-dimensional figure where any cross-section is a circle.

The square root of a positive number \(n\) is the positive number whose square is \(n\). It is also the side length of a square whose area is \(n\). The square root of \(n\) is written as \(\sqrt{n}\).

The square root of 16 is written as \(\sqrt{16}\). Its value is 4 because \(4^2\) is 16.

\(\sqrt{16}\) is also the side length of a square that has an area of 16.

Supplementary angles have measures that add up to 180\(^\circ\).

For example, a \(15^\circ\) angle and a \(165^\circ\) angle are supplementary.

The surface area of a polyhedron is the number of square units that covers all of its faces with no gaps or overlaps.

For example, the 6 faces of a cube each have an area of 9 cm². So, the surface area of the cube is \(6 \boldcdot 9\), or 54 cm².

A system of equations is a set of 2 or more equations. Each equation has 2 or more variables. A solution to the system is values for the variables that make all the equations true.

These equations make up a system of equations:

\(\displaystyle \begin{cases} x + y = \text-2\\x - y = 12\end{cases}\)

The solution to this system is \(x=5\) and \(y=\text-7\). When these values are substituted for \(x\) and \(y\), both equations are true: \(5+(\text-7)=\text-2\) and \(5-(\text-7)=12\).

Terms are the parts of an expression that are added together. They can be a single number, a variable, or a number and a variable that are multiplied together.

• 7, \(y\), and \(9a\) are examples of terms.
• The expression \(5x+3-18\) has 3 terms: \(5x\), 3, and -18.

A tessellation is a repeating pattern of 1 or more shapes. The sides of the shapes fit together with no gaps or overlaps. The pattern goes on forever in all directions.

This diagram shows part of a tessellation.

A transformation is a translation, rotation, reflection, or dilation, or a combination of these.

A translation is a transformation that “slides” a figure along a straight line. Every point on the figure moves a given distance in a given direction.

This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow.

A transversal is a line that crosses parallel lines.

This diagram shows a transversal line \(k\) intersecting parallel lines \(m\) and \(\ell\).

A two-way table shows data for 2 categorical variables. One variable is shown in rows and the other in columns. Each entry is the frequency or relative frequency of the category shown by the column and row headings.

This two-way table shows the results of a study. The study looked at how meditation affects the way athletes feel. 

A variable is a letter that represents a number. Different numbers can be chosen for the value of the variable.

In the expression \(10-x\), the variable is \(x\).

• If the value of \(x\) is 3, then \(10-x=7\), because \(10-3=7\).
• If the value of \(x\) is 6, then \(10-x=4\), because \(10-6=4\).

Vertical angles are opposite angles that share the same vertex. They are formed when two lines cross each other. Their angle measures are equal.

Angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).

Angles \(AED\) and \(BEC\) are another pair of vertical angles.

Volume is the number of cubic units that fill a three-dimensional region with no gaps or overlaps.

This rectangular prism has 3 layers that are each 20 units³. So, the volume of the prism is 60 units³.