# Construction: Bisect Angle

 Definition The bisector of an angle is a ray whose end point is the vertex of the angle and which divides the angle into two equal angles. In the diagram to the right, the ray CD is the bisector of the angle ACB if and only if the angles ACD and BCD have equal measures.

 Bisect Angle. To construct the Angle Bisector of an angle follow the following steps. Given. An angle to bisect. For this example, angle ABC. Step 1. Draw an arc that is centered at the vertex of the angle. This arc can have a radius of any length. However, it must intersect both sides of the angle. We will call these intersection points P and Q This provides a point on each line that is an equal distance from the vertex of the angle. Step 2. Draw two more arcs. The first arc must be centered on one of the two points P or Q. It can have any length radius. The second arc must be centered on whichever point (P or Q) you did NOT choose for the first arc. The radius for the second arc MUST be the same as the first arc. Make sure you make the arcs long enough so that these two arcs intersect in at least one point. We will call this intersection point X. Every intersection point between these arcs (there can be at most 2) will lie on the angle bisector. Step 3. Draw a line that contains both the vertex and X. Since the intersection points and the vertex all lie on the angle bisector, we know that the line which passes through these points must be the angle bisector.

Now, try to do this construction yourself.

## Applet Instructions

• Drawing lines. Start by depressing the ruler button. Then, click on the point where the line should begin. You can then move the mouse to the other point and click again.
• Drawing arcs. Start by depressing the compass button. Click on the center of the arc. Use the up and down arrow keys to increase or decrease the angle of the arc (or use the method listed below). Click again to place the arc.
• Drawing arcs with same radius. If you hold down the "Shift" key when you select the first point of the arc the radius of your new arc will be same as that of last arc drawn.
• Selecting Items. Make sure both the ruler and compass are not depressed. Then, you can select items by clicking on them. The color of the marks will change from red to green. To deselect something click on it again.
• Adjusting lines and arcs. After placing a line or arc you can make adjustments to them. First, selecting the object you want to change. Then, by clicking (not holding down mouse button) on different points you can make different adjustments.
• Lines. Clicking on either endpoint of a line will release that point and thus allow you to move.
• Arcs. When you select an arc four points will be drawn in addition to the arc. By clicking on each of these points you can modify a different aspect of the arc.
1. Center of the circle containing the arc. You can move the center while leaving the center of the curve in the same place. If you hold down the shift key while moving this point, the entire arc will move and keep the same relative position to the center. Be careful not to move the curve off of the screen.
2. Center of the curve. You can move the position of the curve while leaving the center of the circle in the same place. If you hold down the shift key while you are moving, the radius is remain constant.
3. Ends of the curve. You can adjust the length of the curve.
The best way to see what can be adjusted is to just try things.

• Starting Over. To start over click the reset button.
• Checking construction. To check your construction select the "Check" button. A message indicating the first error that is found or that the construction is correct will be shown in the window to the right.