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
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
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.
- Drawing lines. Start by depressing the ruler button. Then,
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
- 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.
The best way to see what can be adjusted is to just try things.
- 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.
- 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.
- 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.
- Ends of the curve. You can adjust the length of the curve.
- 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.