Compass & Straightedge Constructions Cheat Sheet

Key Facts

• A compass preserves distance, so if its opening is set to $\overline{AB}$, any arc made with that opening marks points the same distance from the center.
• A straightedge draws a line through two points, but it is not used to measure length or mark equal distances.
• The perpendicular bisector of $\overline{AB}$ is the line through points where equal-radius arcs from $A$ and $B$ intersect, and every point on it is equidistant from $A$ and $B$.
• To construct the midpoint $M$ of $\overline{AB}$, construct the perpendicular bisector of $\overline{AB}$ and label its intersection with $\overline{AB}$ as $M$, so $AM = MB$.
• An angle bisector of $\angle ABC$ divides it into two congruent angles, so $m\angle ABX = m\angle XBC = \frac{1}{2}m\angle ABC$.
• To copy $\overline{AB}$ from point $P$, draw a ray from $P$, set the compass to $AB$, and mark $Q$ on the ray so $PQ = AB$.
• To construct a line through $P$ perpendicular to line $\ell$, make two equal-distance marks on $\ell$ from $P$ when possible, then construct the perpendicular bisector of the marked segment.
• To construct a line through $P$ parallel to line $\ell$, copy an angle formed by a transversal with $\ell$ at point $P$ so the corresponding angles are congruent.

Vocabulary

Compass

A tool used to draw circles and arcs and to transfer a fixed distance without measuring.

Straightedge

A tool used to draw straight lines through points without using measurement marks.

Perpendicular bisector

A line that intersects a segment at its midpoint and forms right angles with the segment.

Angle bisector

A ray that divides an angle into two congruent angles.

Congruent

Figures, segments, or angles that have the same size and shape, often written using $\cong$.

Arc

Part of a circle drawn with a compass from a fixed center and radius.