Draw an angle. Create a copy of this angle using only a ruler and compass.

Question

Homi · Accepted Answer

Understand the Question

To copy a given angle $\angle BAC$ using only a ruler and compass, we construct an angle $\angle QPS$ of equal measure without measuring the degree value with a protractor:

Draw a reference ray $\overrightarrow{PQ}$ to serve as the base of the new angle.
Use a compass with a fixed radius to mark intersecting arcs on both the original angle arms and the new ray.
Measure the opening distance between the two arms on the original angle and transfer it to the new arc to locate the second arm.

Step 1 · Draw the Given Angle and the Base Ray

Let the given angle be $\angle BAC$ with vertex $A$ .

Draw a new ray $\overrightarrow{PQ}$ with endpoint $P$ , which will serve as the vertex of the copied angle. Diagram 1

Step 2 · Draw an Arc on the Original Angle

With $A$ as centre and any convenient radius, draw an arc intersecting ray $\overrightarrow{AB}$ at point $D$ and ray $\overrightarrow{AC}$ at point $E$ . Diagram 2

Step 3 · Draw a Corresponding Arc on the New Ray

With the same radius and $P$ as centre, draw an arc intersecting ray $\overrightarrow{PQ}$ at point $R$ . Diagram 3

Step 4 · Measure the Width of the Original Angle

Place the compass pointer at $D$ and adjust the opening so the pencil point rests exactly on $E$ . This sets the compass to the distance $DE$ . Diagram 4

Step 5 · Transfer the Width to the New Arc

With $R$ as centre and radius equal to $DE$ , draw an arc intersecting the previous arc at point $S$ . Diagram 5

Step 6 · Draw the Second Ray to Complete the Angle

Draw ray $\overrightarrow{PS}$ passing through point $S$ .

$\angle QPS = \angle BAC$ Diagram 6

Answer

$\angle QPS = \angle BAC$

Common Mistakes

Changing Compass Radius: Altering the compass width between drawing the arc at vertex $A$ and the arc at vertex $P$ will change the scale and result in an incorrect angle.
Incorrect Arc Center: Placing the compass at the vertex $P$ instead of point $R$ when transferring the distance $DE$ .
Inaccurate Width Measurement: Not setting the compass precisely between points $D$ and $E$ , leading to an angle that does not match the original.

Question 15

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