To construct the angle bisector of a given angle, follow these steps:
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Draw the Angle: Start with the angle you want to bisect. Name the vertex of the angle as point A and the two rays as AB and AC.
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Draw an Arc: Place the compass point on the vertex A and draw an arc that intersects both rays (AB and AC). Let's label the points where the arc intersects the rays as points D (on AB) and E (on AC).
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Draw Arcs from D and E: Without changing the radius of the compass, place the compass point on D and draw an arc in the interior of the angle. Then, keeping the same radius, place the compass point on E and draw another arc that intersects the first arc. Label the point where they intersect as F.
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Draw the Angle Bisector: Use a straightedge to draw a line from point A to point F. This line AF is the angle bisector of the original angle ∠BAC.
This procedure results in the angle bisector, which divides the angle into two equal parts. This method is a classic construction used in various geometrical problems and proofs.