copying an angle — Definition, Formula & Examples
Copying an angle is a compass-and-straightedge construction that recreates an angle with the exact same measure at a different location, without using a protractor.
Given an angle and a ray , copying an angle is the classical construction that produces a ray such that , using only an unmarked straightedge and a compass.
How It Works
The method works by transferring a chord length from one angle's arc to another. You draw an arc on the original angle to mark how far apart the two sides are at a fixed radius. Then you draw an arc of the same radius at the new vertex and transfer that chord distance. Because arcs of equal radius with equal chords subtend equal central angles, the new angle is congruent to the original.
Worked Example
Problem: Copy angle ABC (which measures 50°) onto ray DE so that the new angle at D is congruent to angle ABC.
Step 1: Place the compass point on vertex B of the original angle. Draw an arc that crosses both sides of the angle, creating intersection points P (on ray BA) and Q (on ray BC).
Step 2: Without changing the compass width, place the compass point on D (the vertex of the new angle). Draw an arc crossing ray DE, creating intersection point P'.
Step 3: Set the compass width to the distance PQ (the chord between the two intersection points on the original angle).
Step 4: Place the compass on P' and draw a small arc that intersects the arc from Step 2. Label this intersection Q'.
Step 5: Draw ray DF through Q'. Angle FDE is now congruent to angle ABC.
Answer: The constructed angle FDE measures 50°, congruent to the original angle ABC.
Why It Matters
Copying an angle is a foundational skill in geometry proofs and constructions. It is used as a sub-step when constructing parallel lines (alternate interior angles), bisecting angles, and building regular polygons. Mastering it is typically required on high-school geometry exams and standardized tests.
Common Mistakes
Mistake: Changing the compass width between the arc on the original angle and the arc on the new vertex.
Correction: The radius must stay the same for both arcs. Only adjust the compass when you measure the chord distance PQ in the separate step.