New York Geometry - 2020 Edition
Investigation: Construct angle bisectors
Lesson

### Compass and straightedge construction.

Recall that to bisect an angle means to cut the angle in half.  To bisect an angle you will need a compass and a straight edge.

#### Steps for construction

1. Start with $\angle ABC$ABC
2. Place the compass on the vertex $A$A.
3. Adjust the width of the compass to any length
4. Draw an arc on both legs of the angle. Call these intersection points $M$M and $N$N.
5. Move the compass to point $M$M, and draw another arc on the interior of $\angle ABC$ABC
6. Don't change the width of the compass; move to point $N$N and cross the arc with another.
7. Draw a line from the vertex $B$B to this intersection with the straightedge.
8. This line bisects the angle.

### Dynamic software construction

We can also construct an angle bisector using dynamic geometry software.  Press the pause/play button in the applet below to see the steps of the construction in action.  To test the construction, move the points around and see if the angles are always the same size.

Now it's your turn!  Repeat the steps for construction to construct your own angle bisector.  Click here to open the applet in a larger web browser window.

#### Steps for construction using technology

 Move Tool Point Tool Line Tool Line Segment Tool Ray Tool Compass Tool
1. Use ray tool to draw an arbitrary angle $\angle BAC$BAC.
2. Choose the compass tool, then click on $\overline{AB}$AB to set the radius length to be the same as $AB$AB, then click on center $A$A to draw the circle.
3. Use the point tool to label the intersection of circle $A$A and $\overrightarrow{AC}$AC as point $D$D.
4. Select the compass tool. Select points $A$A and $B$B. Then select point $B$B to construct a circle at $B$B with radius $AB$AB.
5. Select the compass tool and construct a circle with radius $AB$AB and center at $D$D.
6. Use the point tool to label the intersection of circles $B$B and $D$D. Call this point $E$E.
7. Use the ray tool to draw a ray from $A$A through $E$E. This is the angle bisector of $\angle BAC$BAC.
8. Use the move tool to drag points and check that your construction holds.