A construction uses only two tools. The compass draws circles and arcs; the straightedge draws lines but has no marks for measuring. Greek geometers chose these tools on purpose: with no ruler or protractor, every result has to come from reasoning, not reading a scale. You will use the angle ideas from Angles and Angle Bisectors.
Why it works
Every construction rests on one fact: all points on an arc are the same distance from its center. So a compass copies a length exactly. Equal lengths then give triangles with three matching sides, and triangles with three matching sides are congruent (SSS, which you will prove in SSS, SAS and ASA). Congruent triangles have matching angles. That is why the steps work, not just how to do them.
Worked example: Copy an angle
Copy $\angle A$ so that the copy opens from a ray that starts at $D$.
- With the compass on $A$, draw an arc that crosses both sides of the angle at $P$ and $Q$.
- Keep the width. With the compass on $D$, draw the same arc, crossing the ray at $R$.
- Set the compass to the distance $PQ$. With it on $R$, mark $S$ where it crosses the arc.
- Draw ray $DS$. Triangles $APQ$ and $DRS$ have $AP = DR$, $AQ = DS$ and $PQ = RS$, so they are congruent by SSS, and $\angle D \cong \angle A$.
Answer: Ray $DS$ makes an angle congruent to $\angle A$.
Worked example: Bisect an angle
Bisect $\angle A$, and explain why the two halves are equal.
- With the compass on $A$, draw an arc crossing the sides at $P$ and $Q$, so $AP = AQ$.
- Open the compass to more than half of $PQ$. Draw an arc from $P$ and, with the same width, an arc from $Q$. They cross at $X$, so $PX = QX$.
- Draw ray $AX$. Triangles $APX$ and $AQX$ share side $AX$, so they are congruent by SSS.
- Matching angles of congruent triangles are equal: $\angle PAX \cong \angle QAX$.
Answer: Ray $AX$ bisects $\angle A$.
To bisect a segment, draw equal arcs from both endpoints above and below it; the line through the two crossings cuts the segment in half at a right angle. You will study it in Constructing Perpendicular and Parallel Lines.
Key idea
A compass copies lengths; a straightedge draws lines. To copy a segment or angle, or to bisect one, use arcs of equal radius, and the result is exact because it creates congruent triangles.
Common mistake
Changing the compass width partway through. The bisector works only because $PX = QX$; if you widen the compass between the two arcs, $PX \ne QX$ and the ray misses the true bisector. Check: keep the compass hinge firm, and if the two halves do not look equal, redraw the arcs.