How to Find the Angle Measurement of Intersecting Lines


Two lines intersect on a graph when they have different slopes. The slope, or gradient, describes the change in a line's y-coordinates with respect to a change in its x-coordinates; and the line, the change in its y-coordinates and the change in its x-coordinates together form a triangle. Within this triangle, that gradient also equals the trigonometric tangent of the angle between the horizontal line and the hypotenuse. Geometry can relate these angles associated with the lines to calculate the separate angle where the lines intersect.

  • Determine each line's gradient from the x coefficient in the associated functions. For instance, if one line has an equation of "y = 2x + 5," its gradient is 2. If the other line has an equation of "y = 5x + 3," its gradient is 5.

  • Identify the line with the smaller gradient. With this example, that would be the "y = 2x + 5" line. Call this the "first line," and call the other one the "second line."

  • Find the inverse tangent of each line's gradient. If you don't have a scientific calculator, use the online one from the first link in "Resources." Tan-1 (2) is 63.4, and tan-1 (5) is 78.7. These are the angles between each line and the x-axis.

  • Subtract the angle you calculated in the previous step for the second line from 180: 180 - 78.7 = 101.3. The two lines and the x-axis form a triangle, and this angle and the other angle you calculated in the previous step are two of this triangle's angles.

  • Subtract the angle you calculated in the previous step and the first angle you calculated in Step 3 from 180: 180 - (101.3 + 63.4) = 15.3. This is the third angle in the triangle the two lines form with the x-axis, which is the angle that the lines' intersection forms.

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