133. Copy That Angle! · Construct a congruent angle using arcs and radii

Try to break it

Drag C around B to change ∠ABC. Watch ∠TPQ track it exactly — the construction copies whatever angle you set. Try to set an angle the arcs can't copy; you can't. The arc-and-compass method never miscopies.

How you build it

Construct a copy of an angle.

• Place point P — the vertex where you will copy angle ABC to.
• Place point Q away from P. Q marks the direction of the base arm ray PQ.
• Ray tool: click P, then click Q. This draws the base arm ray PQ.
• Arc tool: click centre B, drag until the arc snaps to point D. This locks the compass to radius BD — the same radius as the given arc at B.
• Arc tool: click centre P. Radius is locked from step 4. This arc crosses ray PQ — mark that crossing as R next.
• Place point R exactly where the arc from step 5 crosses ray PQ.
• Arc tool: click centre D, drag until the arc snaps to point E. This locks the compass to the chord DE — the exact width of angle ABC captured in the arc.
• Arc tool: click centre R. Compass is locked to chord DE from step 7. This arc crosses the P-arc from step 5 at point S.
• Place point S exactly where the arc from step 8 meets the P-arc from step 5. Ray PS will be the second arm of the copied angle.
• Ray tool: click P, then click S. The angle between ray PQ and ray PS equals angle ABC exactly. The copy is complete!

The proof, step by step

Prove that the constructed angle ∠TPQ equals the given angle ∠ABC.

• In ΔBDE and ΔPRS, BD = PR (radii of equal arcs), BE = PS (radii of equal arcs), and DE = RS (by construction).
• By SSS congruence criterion, ΔBDE ≅ ΔPRS.
• Therefore, ∠DBE = ∠RPS (c.p.c.t.), which means ∠ABC = ∠TPQ.

Worked example

In the construction of a congruent angle, if the given angle measures 75°, what is the measure of the constructed angle?

By the construction of a congruent angle, the constructed angle is always equal to the given angle. So, angle TPQ = 75°.

• 37.5°
• 75° — correct
• 105°
• 150°