Constructions

56. The Angle Twin · Copy any angle perfectly every time

The copied angle ∠PQR is always equal to the original ∠AOB.

To copy an angle: from the original vertex O, draw an arc cutting both arms. Copy that arc with the same radius at the new vertex Q. Set the compass to the chord between the two arc-arm intersections, mark that chord at Q. The new ∠PQR equals the original ∠AOB exactly.

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Selina ICSE: Constructions

What this lesson covers

Try to break it

What if we change ∠AOB? Try dragging B to make it bigger or smaller. See how ∠PQR changes instantly to match?

How you build it

Construct a copy of a given angle.

Mark point O — the vertex of the angle you will copy. Keep it toward the left.
Draw a ray from O — the first arm of the angle, OA.
Draw a second ray from O — the other arm, OB. The opening between the arms is ∠AOB, the angle to copy.
With centre O, draw an arc that cuts across both arms — click O, then click outward to set the opening.
Mark point C where the arc crosses the first arm.
Mark point D where the arc crosses the second arm.
Mark point Q toward the right — the vertex of the copy.
Draw a ray from Q — the base arm of the copy, QR.
With centre Q and the same opening, draw an arc that cuts the base arm.
Mark point T where this arc crosses the base arm.
With centre C, open the compass out to D — this sets the opening to the distance from C to D.
With centre T and that same opening, draw an arc that crosses the first arc near Q.
Mark point P where the two arcs cross. P and Q together give the copied arm.
Draw the ray from Q through P — the second arm. ∠PQR now equals ∠AOB exactly. The angle is copied.

The proof, step by step

Why does it work? Look at triangles OCD and QTS. OC=QT (same radius), OD=QS (same radius), and CD=TS (we copied the distance). By SSS, the triangles are congruent, so the angles are equal!

OC = QT and OD = QS (Radii of equal arcs).
CD = TS (By construction, compass distance copied).
ΔOCD ≅ ΔQTS (SSS Congruence Criterion).
∠AOB = ∠PQR (Corresponding parts of congruent triangles).

Worked example

In the construction of copying an angle ∠AOB to point Q, an arc is drawn with centre O cutting the arms at C and D. Another arc is drawn with centre Q and radius OC. What is the purpose of drawing an arc with centre T and radius equal to CD?

Drawing an arc with centre T and radius CD locates point S such that chord TS equals chord CD. This ensures triangles OCD and QTS are congruent by SSS, guaranteeing that ∠AOB = ∠PQR.

To ensure the new angle is twice the original.
To locate point S such that chord TS equals chord CD, ensuring congruent triangles. — correct
To draw a tangent to the circle.
To bisect the angle.