Constructing a 90-Degree Angle
To construct a 90-degree angle, you will need a compass, a straightedge (ruler), and a pencil. Follow these steps:
1. **Draw a Base Line**
- Start by drawing a horizontal line using the straightedge. This will serve as one arm of your angle.
2. **Locate the Vertex**
- Mark a point on the line. This point will be the vertex of your 90-degree angle.
3. **Create an Arc**
- Place the compass point on the vertex and draw an arc that intersects the base line.
4. **Mark the Intersection**
- Label the point where the arc meets the base line as point A.
5. **Draw a Perpendicular Line**
- Without changing the compass width, place the compass on point A and draw an arc above and below the base line.
6. **Create Two Intersections**
- Mark these intersection points as B (above) and C (below the base line).
7. **Draw the Final Line**
- Using the straightedge, draw a line connecting points B and C. This line is perpendicular to your base line, creating a 90-degree angle.

Dividing the Angle into Four Equal Parts
Now that you have a 90-degree angle, you can divide it into four equal parts:
1. **Set the Compass Width**
- Keep the compass at a medium width and place the point on the vertex.
2. **Draw Arcs**
- Draw an arc that intersects both arms of the angle. Label the intersection points as D and E.
3. **Bisect the Segments**
- Using the compass, place it on point D and draw an arc above and below the angle. Repeat from point E.
4. **Mark the Intersections**
- Label the intersections of the arcs as F and G.
5. **Draw Final Lines**
- Connect the vertex to points F and G to create three additional lines, thus dividing the 90-degree angle into four equal 22.5-degree angles.