Calculating the Angle Formed From Intersecting Lines

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn a different way of finding an angle between two intersecting lines. This method is useful when you know only the slopes of the lines.

Intersecting Lines Angle

All angles are formed from two intersecting lines like this.


intersecting lines angle


As you can see, the lines keep going forever with only a part of each line making up the sides of the angle.

Imagine that you are an architect and you are presented with the picture of the two intersecting lines that you just saw. You need to figure out the angle's measurement so you can write it on your blueprint. You need this angle because it tells you at what angle to build a certain decoration for the mayor of your town. What can you do? The only information you were given is the slopes of the two lines.


intersecting lines angle


Using the Angle of Inclination

Since you are only given the slopes of the lines, you can use what is called the angle of inclination of your lines. The angle of inclination is the angle the line forms with the horizontal. It is given by this formula.


intersecting lines angle


The m is your slope. So, all you need to do is to plug in your slope and then find the arctangent of it. When you do, make sure you note whether you are using degrees or radians.

For your line with a slope of 1/2, your angle of inclination is calculated to be this.


intersecting lines angle


For your line with the slope of 2, your angle of inclination is this.


intersecting lines angle


Finding Your Angle

Once you've found your angles of inclination, you'll subtract them from each other.


intersecting lines angle


If you get a negative number, you'll take the absolute value. In this case, you get 36.87 degrees. This number is less than 90 degrees, so you are looking at the acute angle. Since this is the angle that you want, 36.87 degrees is your answer.

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