Saadiq constructed ∠BAC and an auxiliary segment where he placed point D at one end. With the help of a compass and straightedge, what next step should Saadiq do to construct a copy of ∠BAC ?

Answer 1

To construct a copy of angle \( \angle BAC \) using a compass and straightedge after having placed point \( D \), Saadiq should follow these steps:

1. Draw a Ray: Draw a ray \( DE \) that will be one side of the copied angle.

2. Measure \( \angle BAC \):

   • Place the compass point at point \( A \) (the vertex of \( \angle BAC \)).
   • Open the compass to a convenient width (this width can be any length that doesn't exceed the length of sides \( AB \) or \( AC \), but typically a medium width is chosen).
   • Draw an arc that intersects both rays \( AB \) and \( AC \). Label the points where the arc intersects \( AB \) and \( AC \) as \( F \) and \( G \) respectively.

3. Transfer the Arc:

   • Without changing the compass width, place the compass point at point \( D \).
   • Draw a similar arc that intersects ray \( DE \). Label the point where it intersects as \( H \).

4. Locate Points for the Angle:

   • Measure the distance between points \( F \) and \( G \) using the compass.
   • Keeping the same compass width, place the compass point at point \( H \) and draw an arc to mark a point \( I \) on the arc.

5. Final Steps:

   • Now, you have the measure for creating a point that corresponds to \( G \). The line segment \( HI \) will create the angle \( \angle DHE \) that is congruent to \( \angle BAC \).
   • Finally, use a straightedge to draw the line from \( D \) to \( I \), thus forming \( \angle DHE \) which is a copy of \( \angle BAC \).

With these steps, Saadiq successfully constructs a copy of the angle \( \angle BAC \) at point \( D \).