In this explainer, we will learn how to find the measure of an acute angle between two straight lines in the coordinate plane.

We recall that the equation of any straight line in the coordinate plane can be written in the form
, where , , and are constants. This is called the
**general form** of a straight line. Alternatively, when we know the slope or gradient of a line, , and
its -intercept, , we can write the equation in **slopeβintercept** form:

(Note that the constant, , in the slopeβintercept form and the general form is not the same.)

Given the slope, , of a line and a point, , on the line,
we may also write the equation of the line in **pointβslope** form:

If we have the slope, , of a line, we know that when is positive, the angle, , measured counterclockwise from the positive direction of the -axis to the line is acute. That is, . We also know from the pointβslope form that if we have two distinct points on a line, say and , then the slope of the line is the ratio of the difference in to the difference in :

Now, from a geometrical perspective, if we form a right triangle using our two points and and a third point in the plane as shown in the diagram, then recalling that , we have

Since is the difference in the -coordinates of two points on the line and is the difference in the corresponding -coordinates of two points on the line, this means that

Similarly, if the angle measured counterclockwise from the positive direction of the -axis to the line is obtuse, that is, , as shown in the diagram below, then the slope of the line passing through points and is

Hence, whether the angle, , measured counterclockwise from the positive direction of the -axis to the line is acute or obtuse, the slope of the line, , is equal to .

It is worth mentioning that although we can extend our method to the special cases of vertical and horizontal lines, we do not cover these in this explainer. We note simply that horizontal lines have a slope and an angle of , and vertical lines have an undefined slope and an angle of .

Now, suppose we have two lines in the coordinate plane, with slopes and , for example, as shown in the diagram below. In this case, , and both angles are acute.

Assuming that the lines are not parallel, that is, , and given that the angles in a triangle must sum to , we have

Taking the tangent on both sides then gives us . Now, using the trigonometric identity, we have

This is true for any two lines described as above. However, while the proof differs slightly depending on the position of the lines and the location of their point of intersection, provided neither line is vertical, that is, neither nor , this result holds for any two nonparallel lines with angles and , measured counterclockwise from the positive direction of the -axis, to each line, respectively.

Hence, since and , this leads us to the following definition.

### Definition: The Angle between Two Straight Lines in the Coordinate Plane

An angle between two nonparallel lines in the coordinate plane with slopes and , such that , is given by

If the lines are parallel, then and there is no angle between them. If the lines are neither parallel nor perpendicular, then there are two angles between them. We refer to the smaller angle as βthe angleβ or βthe acute angleβ between the lines.

A negative tangent corresponds to the larger, obtuse angle, , and to ensure that the tangent is that of the acute angle, , we take the absolute value. Hence, the tangent of the acute angle between two lines in the coordinate plane is

Note that if , then the denominator is equal to 0, and this expression is undefined. In this case, the lines are perpendicular, and .

Letβs see how this works in an example where we are given the slopes of two lines.

### Example 1: Finding the Measure of the Angle between Two Straight Lines given Their Slopes

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of 5 and .

### Answer

Knowing the slopes, and , of two lines in the coordinate plane, we can find the acute angle, , between the lines using the formula

Letting and , we have

Evaluating the right-hand side then leads to and taking the inverse tangent on both sides gives us

We are asked to find the measure of the angle to the nearest second, and to do this, we recall that there are 60 minutes in one degree and 60 seconds in one minute. We therefore multiply the decimal part of our degrees by 60: . Hence, we have (minutes), and multiplying the decimal part of our minutes by 60 gives us (seconds).

The measure of the acute angle between the two lines, to the nearest second, is therefore .

Our next example demonstrates how to find the angle between two lines in the coordinate plane, where the lines are both given in general form.

### Example 2: Finding the Angle between Two Straight Lines in Two Dimensions

Find the measure of the acute angle between the two straight lines whose equations are and to the nearest second.

### Answer

We are given two lines whose equations are in general form, that is, in the form , where , , and are real numbers. To find the acute angle, , between the two lines, we will use the formula where and are the slopes of the given lines. We will therefore, need to find these two slopes given in each case by .

Our lines are whose slopes are therefore and .

Substituting these two values into the formula for , we then have

Now, taking the inverse tangent of both sides, we find

We are asked for the measure of the angle to the nearest second. To find this, we recall that there are 60 minutes in one degree and 60 seconds in one minute. We therefore multiply the decimal part of our degrees by 60: . Hence, we have (minutes), and multiplying the decimal part of our minutes by 60 gives us (seconds).

The measure of the acute angle between the two lines, to the nearest second, is therefore, .

In our next example, we find the acute angle between one line whose equation is given in general form and a second line passing through two known points.

### Example 3: Finding the Angle between Two Straight Lines in Two Dimensions

Determine the measure of the acute angle between the straight line and the straight line passing through the points and to the nearest second.

### Answer

To find the acute angle, , between the two given lines, we can use the formula where is the slope of our first line, which we will denote , and is the slope of the second line, which we will denote . To use this formula, we will need to find the slopes of the two lines.

Our first line is given in the general form: , where , , and are real numbers and where, in our case, , , and . The slope, , is given by ; hence, .

To find our second slope, , we recall that, given two distinct points on a line, and , the slope of the line through the points is given by the change in divided by the change in . That is,

For our second line, , we are given the coordinates of two points on the line, and . The slope, , of is therefore

Now that we have the two slopes and , we can use the stated formula for the tangent of the acute angle, , between the two lines:

Now, taking the inverse tangent, we find

We are asked for the angle to the nearest second. To find this, we recall that there are 60 minutes in one degree and 60 seconds in one minute. We therefore multiply the decimal part of our degrees by 60: . Hence, we have (minutes), and multiplying the decimal part of our minutes by 60 gives us (seconds).

The measure of the angle between the two lines, to the nearest second, is therefore .

Up till now, our lines have been defined either in general form or via known points on the line from which we were able to find the slopes. However, there are other forms in which straight lines can be expressed, as in the following definition.

### Definition: The Vector, Parametric, and Cartesian Forms of a Line in the Coordinate Plane

A line passing through a point A with position vector , in the direction of the vector , can be written in the following forms:

In the vector form, each unique value of the real parameter gives the position vector of a point on the line, and in the Cartesian form, we assume both and are nonzero.

Note that by solving each of the parametric equations for and equating, we obtain the Cartesian form, which can be rearranged as follows:

This is now in slopeβintercept form, , where here the slope . Hence, given a line in any of the above forms and, in particular, its direction vector, we are able to find its slope, which is , provided is nonzero.

In our next example, we will use this to find the angle between two lines whose equations are given in vector and parametric forms.

### Example 4: Finding the Angle between Two Straight Lines in Two Dimensions

Find the measure of the acute angle between the two straight lines and whose equations are and , , respectively, in terms of degrees, minutes, and seconds, to the nearest second.

### Answer

To find the acute angle, , between two lines in the coordinate plane, we use the formula where and are the slopes of the two lines. We must therefore find the slopes of the two given lines.

The first of our two lines, , is given in vector form, that is, in the form where the line passes through the point with position vector , in the direction of the vector . The slope, , of a line with direction vector is .

For our line, , we see that the constant corresponds to the parameter so that our direction vector is . The slope, , of this line is therefore

Our second line, , is given in parametric form. That is, in the form where again the line passes through the point with position vector , in the direction of the vector . Comparing, we see that parameter in corresponds to parameter in the general parametric equations and that our direction vector is therefore . Hence, the slope of line is given by

We can now use our two slopes, and , in the formula to find the acute angle between the two lines:

Now, taking the inverse tangent on both sides, we have

We are asked for the angle to the nearest second. To find this we recall that there are 60 minutes in one degree and 60 seconds in one minute. We therefore multiply the decimal part of our degrees by 60: . Hence, we have (minutes), and multiplying the decimal part of our minutes by 60 gives us (seconds).

Therefore, the measure of the acute angle between the two lines and , to the nearest second, is .

We will use the formula for the tangent of the angle between two lines, in our next example, to find the equations of the lines.

### Example 5: Finding the Equations of Two Straight Lines in Two Dimensions Using the Tangent of the Angle between Them

Let be the angle between two lines that pass through . If and the slopes of the lines are and , with , find the equations of these lines.

### Answer

Recalling that by the angle between two lines, we mean the smaller of the two angles, we are told that two lines, say and , both pass through the point and that the tangent of the angle, , between them is equal to . We are also given that the slope of is equal to and that the slope of is equal to , with .

To find the equations of lines and , we first use the information we have in the formula for the tangent of the angle between two lines to find any possible values of . We can then use the given point and the pointβslope form of a line to find the equations of lines and .

Recall that the formula for the tangent of the acute angle, , between two lines is as follows: where and are the slopes of the two lines. Substituting our values for , , and , we have noting that since is positive, the right-hand side is positive. Rearranging this results in the quadratic equation which we can solve for . Using the quadratic formula or otherwise, we find that there are two solutions: and .

The pointβslope form of a line is where is the slope, and the line passes through the point . Using this with each of our solutions for , together with the point through which both lines pass, will produce equations for and .

Beginning with our solution, , we recall that the slope of is and the slope of . We therefore have

Now with the second solution, , we have that the slope of is and the slope of .

Therefore, in this case, we have

Hence, there are two possible pairs for the equations of the lines: or

We complete this explainer by reminding ourselves of some of the key points.

### Key Points

- To find the acute angle, , between two straight lines with slopes and in the coordinate plane, we use the formula
- If , then the denominator in the formula for is equal to 0, and the expression is undefined. In this case, the lines are perpendicular, and .
- We refer to the acute angle between two lines in the coordinate plane as the angle between the lines.
- If the lines are parallel, they do not intersect, so there is no angle between them.