Textbook

1. Word Knowledge

2. Mathematics Knowledge

2.1 Foundational math

2.2 Pre-algebra

2.3 Algebra I

2.4 Geometry basics

2.4.1 Lines and slopes

2.4.2 Angles and transversals

2.4.3 Basic shapes and area

2.4.4 Circles, polygons, and coordinate geometry

2.5 Algebra II

2.6 Advanced geometry

2.7 Math strategies

3. Paragraph Comprehension

4. Arithmetic Reasoning

5. General Science

6. Mechanical Comprehension

7. Electronics Information

8. Shop Information

9. Auto Information

10. Assembling Objects

Wrapping up

2.4.1 Lines and slopes

Achievable ASVAB

2. Mathematics Knowledge

2.4. Geometry basics

Lines and slopes

7 min read

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Lines

Definitions

Line: A perfectly straight path that extends infinitely in both directions
Line segment: A part of a line with two endpoints (a defined start and end point)

Arrows show that a line continues forever; dots mark a segment’s endpoints.

$A line, with an arrow on each end, and a line segment, with a dot at each end.$

Types of lines

Achievable

A road that continues in both directions can be modeled as a line, while a fence between two posts is a line segment because it has endpoints.

Slopes

Definitions

Slope: A measure of both the steepness and direction of a line, calculated as the vertical change divided by the horizontal change ( $\frac{r i se}{r u n}$ )

Slope is read from left to right.

A helpful way to think about slope is “rise over run”:

Rise = How much the line goes up or down
Run = How much the line goes left or right

This model shows how slope works:

A line rising to the right through two points. A horizontal run and a vertical rise between the points form a right triangle with the line. Slope equals rise over run. — Slope definition

Sidenote

Horizontal and vertical lines

A perfectly horizontal line has a slope of $0$ because it has no rise. Its rise is $0$ , so its slope is always $\frac{0}{a}$ . No matter how large the run is, $0$ divided by any nonzero number is $0$ .

A perfectly vertical line has an undefined slope because it has no run. Its run is $0$ , and division by $0$ is undefined.

Types of slope

Lines can have different types of slope depending on how they move from left to right.

Positive slope: the line goes up as you move to the right
Negative slope: the line goes down as you move to the right
Zero slope: the line is horizontal (no rise)
Undefined slope: the line is vertical (no run)

Four lines: positive slope, rising to the right; negative slope, falling to the right; zero slope, horizontal; and undefined slope, vertical. — Types of slope

Knowledge check: A line goes down as you move from left to right. What type of slope does it have?

(spoiler)

A negative slope.

Interactions of lines

When comparing two lines, it’s helpful to look at their slopes. The slope tells you whether lines will meet, stay apart, or overlap.

Two lines can interact in one of three ways:

They can intersect at exactly one point
They can never intersect
They can overlap and intersect at infinite points.

$Three panels: two lines that cross at one point, two parallel lines that never cross, and two lines that lie on top of each other and share every point.$

Intersection of lines

Achievable

Lines intersect when they have different slope values. Since they rise or fall at different rates, they will meet at some point.
Lines never intersect when they have the same slope but different positions (different y-intercepts). Since they travel at the same rate but sit in different positions, they stay a constant distance apart. These are also called parallel lines. Two different vertical lines, such as $x = 1$ and $x = 4$ , are parallel too, though neither has a slope or a $y$ -intercept.
Lines intersect infinitely when they have the same slope and the same position. Because they sit in the same position and travel at the exact same rate, they are the same line.

Two parallel lines, each marked with matching arrows, that never meet. — Parallel lines

Perpendicular lines

Definitions

Perpendicular lines: Lines that meet at exactly one point and form four $9 0^{\circ}$ angles (right angles). Their slopes are opposite reciprocals of each other.

One important type of intersecting lines is perpendicular lines.

Two lines crossing at a right angle, marked with a small square. — Perpendicular lines

You can also recognize perpendicular lines by looking at their slopes. The slopes of perpendicular lines are opposite reciprocals.

That means:

The signs are opposite, and
The fractions are flipped (reciprocal).

Think: flip the fraction and change the sign.

So if one line has a slope of $+ \frac{2}{3}$ , the line perpendicular to it has a slope of $- \frac{3}{2}$ .

One exception: a horizontal line (slope $0$ ) and a vertical line (undefined slope) are also perpendicular, even though $0$ has no reciprocal.

Two lines through the origin on a coordinate grid with equal scales, y = (2/3)x and y = −(3/2)x. Their slopes are opposite reciprocals, and they meet at a right angle. — Perpendicular line graph

In the graph above, the two lines are perpendicular because their slopes are opposite reciprocals.

Knowledge check: A line has a slope of $- 5$ . What is the slope of a line perpendicular to it?

(spoiler)

Rewrite $- 5$ as $- \frac{5}{1}$ , then flip the fraction and change the sign:

$\frac{1}{5}$

Lines vs. line segments

Line: perfectly straight, extends infinitely both directions (shown with arrows)
Line segment: straight path with two defined endpoints (shown with dots)

Slope basics

Slope = steepness/direction = $\frac{r i se}{r u n}$ , read left to right
Horizontal line: slope = $0$ (no rise, $\frac{0}{a} = 0$ )
Vertical line: slope undefined (no run, division by $0$ )

Types of slope

Positive: line rises left to right
Negative: line falls left to right
Zero: horizontal line
Undefined: vertical line

How lines interact

Different slopes → intersect at exactly one point
Same slope, different y-intercept → never intersect (parallel lines); two different vertical lines (e.g., $x = 1$ and $x = 4$ ) are parallel too
Same slope and same position → same line, infinite intersection points

Perpendicular lines

Meet at one point, form four $90°$ right angles
Slopes are opposite reciprocals: flip the fraction and change the sign
- e.g., slope $+ \frac{2}{3}$ ↔ perpendicular slope $- \frac{3}{2}$
Exception: horizontal (slope $0$ ) and vertical (undefined slope) lines are perpendicular despite no reciprocal relationship

Next | 2.4.2 Angles and transversals

Lines and slopes

Lines

Definitions

Line: A perfectly straight path that extends infinitely in both directions
Line segment: A part of a line with two endpoints (a defined start and end point)

Arrows show that a line continues forever; dots mark a segment’s endpoints.

A road that continues in both directions can be modeled as a line, while a fence between two posts is a line segment because it has endpoints.

Slopes

Definitions

Slope: A measure of both the steepness and direction of a line, calculated as the vertical change divided by the horizontal change ( $\frac{r i se}{r u n}$ )

Slope is read from left to right.

A helpful way to think about slope is “rise over run”:

Rise = How much the line goes up or down
Run = How much the line goes left or right

This model shows how slope works:

Sidenote

Horizontal and vertical lines

A perfectly vertical line has an undefined slope because it has no run. Its run is $0$ , and division by $0$ is undefined.

Types of slope

Lines can have different types of slope depending on how they move from left to right.

Positive slope: the line goes up as you move to the right
Negative slope: the line goes down as you move to the right
Zero slope: the line is horizontal (no rise)
Undefined slope: the line is vertical (no run)

Knowledge check: A line goes down as you move from left to right. What type of slope does it have?

(spoiler)

A negative slope.

Interactions of lines

When comparing two lines, it’s helpful to look at their slopes. The slope tells you whether lines will meet, stay apart, or overlap.

Two lines can interact in one of three ways:

They can intersect at exactly one point
They can never intersect
They can overlap and intersect at infinite points.

Lines intersect when they have different slope values. Since they rise or fall at different rates, they will meet at some point.
Lines never intersect when they have the same slope but different positions (different y-intercepts). Since they travel at the same rate but sit in different positions, they stay a constant distance apart. These are also called parallel lines. Two different vertical lines, such as $x = 1$ and $x = 4$ , are parallel too, though neither has a slope or a $y$ -intercept.
Lines intersect infinitely when they have the same slope and the same position. Because they sit in the same position and travel at the exact same rate, they are the same line.

Perpendicular lines

Definitions

Perpendicular lines: Lines that meet at exactly one point and form four $9 0^{\circ}$ angles (right angles). Their slopes are opposite reciprocals of each other.

One important type of intersecting lines is perpendicular lines.

You can also recognize perpendicular lines by looking at their slopes. The slopes of perpendicular lines are opposite reciprocals.

That means:

The signs are opposite, and
The fractions are flipped (reciprocal).

Think: flip the fraction and change the sign.

So if one line has a slope of $+ \frac{2}{3}$ , the line perpendicular to it has a slope of $- \frac{3}{2}$ .

One exception: a horizontal line (slope $0$ ) and a vertical line (undefined slope) are also perpendicular, even though $0$ has no reciprocal.

In the graph above, the two lines are perpendicular because their slopes are opposite reciprocals.

Knowledge check: A line has a slope of $- 5$ . What is the slope of a line perpendicular to it?

(spoiler)

Rewrite $- 5$ as $- \frac{5}{1}$ , then flip the fraction and change the sign:

$\frac{1}{5}$

Achievable ASVAB

2. Mathematics Knowledge

2.4. Geometry basics

Lines and slopes

7 min read

Font

Discuss

Feedback

Lines

Definitions

Line: A perfectly straight path that extends infinitely in both directions
Line segment: A part of a line with two endpoints (a defined start and end point)

Arrows show that a line continues forever; dots mark a segment’s endpoints.

$A line, with an arrow on each end, and a line segment, with a dot at each end.$

Types of lines

Achievable

A road that continues in both directions can be modeled as a line, while a fence between two posts is a line segment because it has endpoints.

Slopes

Definitions

Slope: A measure of both the steepness and direction of a line, calculated as the vertical change divided by the horizontal change ( $\frac{r i se}{r u n}$ )

Slope is read from left to right.

A helpful way to think about slope is “rise over run”:

Rise = How much the line goes up or down
Run = How much the line goes left or right

This model shows how slope works:

Sidenote

Horizontal and vertical lines

A perfectly vertical line has an undefined slope because it has no run. Its run is $0$ , and division by $0$ is undefined.

Types of slope

Lines can have different types of slope depending on how they move from left to right.

Positive slope: the line goes up as you move to the right
Negative slope: the line goes down as you move to the right
Zero slope: the line is horizontal (no rise)
Undefined slope: the line is vertical (no run)

Knowledge check: A line goes down as you move from left to right. What type of slope does it have?

(spoiler)

A negative slope.

Interactions of lines

When comparing two lines, it’s helpful to look at their slopes. The slope tells you whether lines will meet, stay apart, or overlap.

Two lines can interact in one of three ways:

They can intersect at exactly one point
They can never intersect
They can overlap and intersect at infinite points.

$Three panels: two lines that cross at one point, two parallel lines that never cross, and two lines that lie on top of each other and share every point.$

Intersection of lines

Achievable

Lines intersect when they have different slope values. Since they rise or fall at different rates, they will meet at some point.
Lines never intersect when they have the same slope but different positions (different y-intercepts). Since they travel at the same rate but sit in different positions, they stay a constant distance apart. These are also called parallel lines. Two different vertical lines, such as $x = 1$ and $x = 4$ , are parallel too, though neither has a slope or a $y$ -intercept.
Lines intersect infinitely when they have the same slope and the same position. Because they sit in the same position and travel at the exact same rate, they are the same line.

Perpendicular lines

Definitions

Perpendicular lines: Lines that meet at exactly one point and form four $9 0^{\circ}$ angles (right angles). Their slopes are opposite reciprocals of each other.

One important type of intersecting lines is perpendicular lines.

You can also recognize perpendicular lines by looking at their slopes. The slopes of perpendicular lines are opposite reciprocals.

That means:

The signs are opposite, and
The fractions are flipped (reciprocal).

Think: flip the fraction and change the sign.

So if one line has a slope of $+ \frac{2}{3}$ , the line perpendicular to it has a slope of $- \frac{3}{2}$ .

One exception: a horizontal line (slope $0$ ) and a vertical line (undefined slope) are also perpendicular, even though $0$ has no reciprocal.

In the graph above, the two lines are perpendicular because their slopes are opposite reciprocals.

Knowledge check: A line has a slope of $- 5$ . What is the slope of a line perpendicular to it?

(spoiler)

Rewrite $- 5$ as $- \frac{5}{1}$ , then flip the fraction and change the sign:

$\frac{1}{5}$

Lines vs. line segments

Line: perfectly straight, extends infinitely both directions (shown with arrows)
Line segment: straight path with two defined endpoints (shown with dots)

Slope basics

Slope = steepness/direction = $\frac{r i se}{r u n}$ , read left to right
Horizontal line: slope = $0$ (no rise, $\frac{0}{a} = 0$ )
Vertical line: slope undefined (no run, division by $0$ )

Types of slope

Positive: line rises left to right
Negative: line falls left to right
Zero: horizontal line
Undefined: vertical line

How lines interact

Different slopes → intersect at exactly one point
Same slope, different y-intercept → never intersect (parallel lines); two different vertical lines (e.g., $x = 1$ and $x = 4$ ) are parallel too
Same slope and same position → same line, infinite intersection points

Perpendicular lines

Meet at one point, form four $90°$ right angles
Slopes are opposite reciprocals: flip the fraction and change the sign
- e.g., slope $+ \frac{2}{3}$ ↔ perpendicular slope $- \frac{3}{2}$
Exception: horizontal (slope $0$ ) and vertical (undefined slope) lines are perpendicular despite no reciprocal relationship

Next | 2.4.2 Angles and transversals

Lines and slopes

Lines

Definitions

Line: A perfectly straight path that extends infinitely in both directions
Line segment: A part of a line with two endpoints (a defined start and end point)

Arrows show that a line continues forever; dots mark a segment’s endpoints.

A road that continues in both directions can be modeled as a line, while a fence between two posts is a line segment because it has endpoints.

Slopes

Definitions

Slope: A measure of both the steepness and direction of a line, calculated as the vertical change divided by the horizontal change ( $\frac{r i se}{r u n}$ )

Slope is read from left to right.

A helpful way to think about slope is “rise over run”:

Rise = How much the line goes up or down
Run = How much the line goes left or right

This model shows how slope works:

Sidenote

Horizontal and vertical lines

A perfectly vertical line has an undefined slope because it has no run. Its run is $0$ , and division by $0$ is undefined.

Types of slope

Lines can have different types of slope depending on how they move from left to right.

Positive slope: the line goes up as you move to the right
Negative slope: the line goes down as you move to the right
Zero slope: the line is horizontal (no rise)
Undefined slope: the line is vertical (no run)

Knowledge check: A line goes down as you move from left to right. What type of slope does it have?

(spoiler)

A negative slope.

Interactions of lines

When comparing two lines, it’s helpful to look at their slopes. The slope tells you whether lines will meet, stay apart, or overlap.

Two lines can interact in one of three ways:

They can intersect at exactly one point
They can never intersect
They can overlap and intersect at infinite points.

Lines intersect when they have different slope values. Since they rise or fall at different rates, they will meet at some point.
Lines never intersect when they have the same slope but different positions (different y-intercepts). Since they travel at the same rate but sit in different positions, they stay a constant distance apart. These are also called parallel lines. Two different vertical lines, such as $x = 1$ and $x = 4$ , are parallel too, though neither has a slope or a $y$ -intercept.
Lines intersect infinitely when they have the same slope and the same position. Because they sit in the same position and travel at the exact same rate, they are the same line.

Perpendicular lines

Definitions

Perpendicular lines: Lines that meet at exactly one point and form four $9 0^{\circ}$ angles (right angles). Their slopes are opposite reciprocals of each other.

One important type of intersecting lines is perpendicular lines.

You can also recognize perpendicular lines by looking at their slopes. The slopes of perpendicular lines are opposite reciprocals.

That means:

The signs are opposite, and
The fractions are flipped (reciprocal).

Think: flip the fraction and change the sign.

So if one line has a slope of $+ \frac{2}{3}$ , the line perpendicular to it has a slope of $- \frac{3}{2}$ .

One exception: a horizontal line (slope $0$ ) and a vertical line (undefined slope) are also perpendicular, even though $0$ has no reciprocal.

In the graph above, the two lines are perpendicular because their slopes are opposite reciprocals.

Knowledge check: A line has a slope of $- 5$ . What is the slope of a line perpendicular to it?

(spoiler)

Rewrite $- 5$ as $- \frac{5}{1}$ , then flip the fraction and change the sign:

$\frac{1}{5}$