Drawing Angles in Standard Position

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  • 0:00 Angle Positions
  • 1:03 Standard Position for Angles
  • 1:47 Practice Using Degrees
  • 2:40 Practice Using Radians
  • 4:29 Lesson Summary
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Lesson Transcript
Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Angles can be drawn facing any direction you want, but when you put them in standard position, it's easier to estimate the angle measure and work with them using graphs and coordinates. In this lesson, we'll explore drawing angles in standard position.

Angle Positions

You can draw angles in any position without affecting their measure. You can turn them upside down or have them look like a greater-than sign or less-than sign, any position you like. However, in angular geometry there is one position that's considered the standard position for angles. You see, angular geometry is all about rotation around a circle, and a number of degrees (or radians) will tell you how far you've rotated from a starting point.

Say that you're out in the woods wandering around and someone says 'go east.' You need to know which way east is, so you grab your compass, which tells you which way north is. From there, you know to turn 45 degrees to the right to go straight east. Angles are the same way. You start at one point, the origin, then measure from there. That's what standard position for angles is all about: measuring from a pre-defined starting point.

Standard Position for Angles

So what exactly does the standard position for angles look like? On the Cartesian Coordinate Plane, standard position for angles is when the vertex of the angle sits at the origin of the x-axis and y-axis, and one side of the angle starts at the origin and extends to the right along the x-axis. There's a special term for that side of the angle along the x-axis. The initial side is the side of the angle that starts at the origin and extends to the right along the x-axis between Quadrant I and Quadrant III. To complete the angle, we need a terminal side, which also starts at the origin and extends in a direction counter-clockwise from the initial side.

Practice Using Degrees

Let's try our hand at drawing an angle in standard position measured in degrees. Remember, there are 360° in a full circle. You start with the x- and y-axes, which cross at a right (90°) angle. The initial side of your angle will start at the origin, the point where the axes cross, and then go straight to the right.

Now comes the fun part. You have to figure out how far to rotate around the origin to reach the measure that you need. An excellent way to do this accurately is to use a protractor. Let's say your angle is 30°. Line up the baseline of your protractor with the initial line of your angle, making sure that the origin is centered on the protractor. Now follow the curve on the protractor until you get to the 30° point. Once you connect that mark to the origin, you've drawn a 30° angle in standard position.

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