How to Construct a 75-Degree Angle with Compass and Straightedge

Constructing a 75-degree angle using basic tools such as a compass and a straightedge can be an engaging and educational task. This method is not only useful for geometric practice but also for understanding the principles of angle bisection and construction. In this guide, we will walk you through the step-by-step process to create a perfect 75-degree angle.

Materials Needed

Compass Straightedge ruler without measurements Pencil Paper

Steps to Construct a 75-Degree Angle

1. Draw the Base Line

To start, use your straightedge and pencil to draw a horizontal line. Label this line AB. This line will serve as the base for your construction.

2. Mark a Point

Choose a point on the line AB, label it point A. This point will serve as the vertex of your angle.

3. Construct a 60-Degree Angle

Place the compass point on point A and draw a circle that intercepts line AB. Label the intersection point on the right side as point C. Without changing the compass width, place the compass point on point C and draw an arc above line AB. Keeping the same compass width, place the compass point on point A and draw another arc that intersects the first arc. Label this intersection point as point D. Draw a straight line from point A through point D. This line forms a 60-degree angle with line AB.

4. Construct a 30-Degree Angle

Now, we need to construct a 30-degree angle. Using the compass, place the point on D and draw an arc that intersects the circle again. Label this new intersection point as point E. Draw a straight line from point A through point E. This line forms a 30-degree angle with line AB.

5. Combine the Angles

The angles formed by line AD (60 degrees) and line AE (30 degrees) combine to form a 90-degree angle (60 30). To find a 75-degree angle, we need to bisect this 90-degree angle.

Place the compass point on point A and draw a small arc that intersects both lines AD and AE. Label these intersection points as F and G. Without changing the compass width, place the compass point on F and draw an arc. Then without changing the compass width, place the compass point on G and draw another arc that intersects the first arc. Label the intersection of these arcs as H. Draw a line from point A through point H. This line forms a 75-degree angle with line AB.

Summary

By constructing a 60-degree angle and a 30-degree angle, and then bisecting the angle between them, you can create a 75-degree angle. This method relies on geometric principles and the ability to accurately bisect angles.

Understanding the steps and geometric principles behind angle construction can enhance your skills in various fields, from architecture to design. Practice is key to mastering these techniques. Happy constructing!