Which of the following angles cannot be constructed using a protractor, a compass, and a sharpened pencil?

The key concept here is the difference between angles that can be constructed using only a compass and straightedge (with a pencil) and those that cannot. While a protractor allows you to measure any angle, classical geometric constructions refer to those made without one—just the compass and straightedge.

Constructible angles are angles that can be achieved by basic constructions, such as:

• Bisecting a line or angle (dividing exactly in half)
• Constructing perpendiculars and parallels
• Creating regular polygons like equilateral triangles, squares, and hexagons

Some important angles that can always be constructed include:

• 60° (angle of an equilateral triangle)
• 90° (right angle)
• Other simple angles like 30° (because 60° can be bisected)
• 135° is also constructible, because it is the sum of 90° and 45°. Since both 90° and 45° can be constructed (45° by bisecting 90°), 135° can be constructed as well.

The angle 145° is not constructible with only a compass and straightedge. This is because the constructible angles are generally those that can be obtained by repeatedly bisecting 90° or 60°, adding and subtracting these and their halves (so you get 15°, 30°, 45°, 75°, 120°, etc.). 145° does not fit into this system with only those tools.

In summary, while a protractor allows you to measure and draw any angle, only certain angles can be constructed using the classical geometric tools of a compass and a straightedge. 90°, 135°, and 60° are all among these constructible angles, but 145° is not.