Constructing 30- and 45-Degree Angles — Practice Geometry Questions

By Allen Ma, Amber Kuang

When you deal with geometry problems where you have to construct 30- and 45-degree angles, you may need to do more than one construction to create what the problem is asking for.

The following practice questions ask you to apply your knowledge of constructions to some creative problems.

Practice questions

  1. Construct a 30-degree angle.

  2. Construct a 45-degree angle.

Answers and explanations

  1. Here is the solution:

    image0.png

    To construct a 30-degree angle, you first need to construct an equilateral triangle, which will have three 60-degree angles. To do this, draw a line segment and label it

    image1.png

    Using your compass, measure the length of

    image2.png

    Without changing the width of the compass, place your compass at Point P and draw an arc above

    image3.png

    Repeat this step with your compass at Point Q. The intersection of these arcs is the third vertex of the equilateral triangle. Label that point R and connect the points.

    image4.png

    You now have three 60-degree angles. You can bisect any one of those angles to create a 30-degree angle. Put your compass point at P and draw an arc through the angle. Place the compass point at both locations where the arc intersects the angle and draw an arc each time. Label the intersection of the arcs S. Connect Point S to Point P.

    image5.png

    are both 30-degree angles.

  2. Here is the solution:

    image6.png

    Draw a line segment to be used as a part of the angle; call it

    image7.png

    You first need to create a right angle. To do so, draw a perpendicular bisector to

    image8.png

    Place the compass point on A and open the width of the compass a little more than halfway through the line. Draw arcs above and below

    image9.png

    Using the same compass width, place the compass point at B and draw arcs above and below

    image10.png

    Connect the intersection points of both pairs of arcs. Let C be the point where the perpendicular bisector intersects

    image11.png

    This means that

    image12.png

    is a 90-degree angle.

    To create a 45-degree angle, you need to bisect

    image13.png

    To do so, place your compass point at C and draw an arc through

    image14.png

    Using the same compass width, place your compass point at F and draw an arc in the interior of

    image15.png

    Do the same at Point G. Connect Point C to the intersection of the two arcs. Both

    image16.png

    are 45-degree angles.