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Points and are each at the centers of circles of radius .
Here is a line with a point labeled . Use straightedge and compass moves to construct a line perpendicular to that goes through .
Here is an angle:
For each angle that you draw, explain to your partner how each straightedge and compass move helps you to bisect it.
For each angle that your partner draws, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
We can construct a line that is perpendicular to a given line. We can also bisect a given angle using only a straightedge and compass. The line that goes through the vertex of an angle to divide it into two equal angles is called the angle bisector. Both constructions use 2 circles that go through each other’s centers:
To construct a line perpendicular to line that goes through a given point , start by finding 2 points, labeled here as and , on the given line that are the same distance from . Then create 2 circles of the same size centered at and that go through each other’s centers. Connect the intersection points of those circles to draw a perpendicular line, .
To construct an angle bisector, start by finding 2 points, labeled here as and , that are on the rays and the same distance from the vertex. Then create the 2 circles of the same size centered at and that go through each other’s centers. Connect the intersection points of those circles to draw the angle bisector.
One large circle with a center point and a central angle. Two small circles centered on the intersection of central angle and large circle. A line is drawn from the intersection point of the two small circles through the center of the large circle.
In fact, we can think of creating a perpendicular line as bisecting a 180 degree angle!
An angle bisector is a line through the vertex of an angle that divides it into two congruent angles.
In this diagram, the dashed line is the angle bisector.