Constructions with Compass & Straightedge Cheat Sheet

A printable reference covering copying segments and angles, perpendicular and angle bisectors, parallel lines, triangles, and construction accuracy for grades 8-10.

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Compass and straightedge constructions show how geometric figures can be made using only a compass, an unmarked straightedge, and logical steps. This cheat sheet helps students remember the standard construction moves used in grades 8 through 10 geometry. It is useful for copying measures, building perpendiculars and parallels, constructing triangles, and explaining why a construction works. The most important idea is that the compass transfers equal distances, while the straightedge draws lines through points. Many constructions depend on congruent circles, equal radii, and intersection points that are the same distance from two endpoints. Core results include perpendicular bisectors, angle bisectors, copied angles, and triangle constructions using side lengths and angle measures.

Key Facts

A compass copies a segment by setting its width to $\overline{AB}$ and marking the same radius from a new point, so the new segment has length $AB$ .
A perpendicular bisector of $\overline{AB}$ is made from two equal-radius arcs centered at $A$ and $B$ , and every point on it is equidistant from $A$ and $B$ .
An angle bisector divides $\angle ABC$ into two congruent angles, so $m\angle ABD = m\angle DBC = \frac{1}{2}m\angle ABC$ .
To copy an angle, draw equal-radius arcs from each vertex, copy the arc chord length, and connect the new vertex to the copied intersection point.
A line perpendicular to a given line through a point creates right angles, so each angle formed has measure $90^\circ$ .
A parallel line through a point can be constructed by copying a corresponding angle, because congruent corresponding angles imply parallel lines.
A triangle can be constructed by $SSS$ when the side lengths satisfy the triangle inequality $a + b > c$ , $a + c > b$ , and $b + c > a$ .
Construction marks should stay visible because arcs and intersection points provide evidence that lengths or angles are congruent.

Vocabulary

Compass: A tool used to draw circles and arcs and to transfer equal distances without measuring with a ruler.
Straightedge: An unmarked tool used to draw a straight line through two points.
Arc: A curved part of a circle drawn by a compass using a fixed center and radius.
Perpendicular Bisector: A line that crosses a segment at its midpoint and forms angles of $90^\circ$ with the segment.
Angle Bisector: A ray that divides an angle into two congruent angles of equal measure.
Congruent: Figures or measures are congruent when they have exactly the same size and shape, such as $\overline{AB} \cong \overline{CD}$ .

Common Mistakes to Avoid

Changing the compass width while copying a segment is wrong because the copied length will no longer equal the original length.
Using a marked ruler to measure instead of constructing is wrong because compass and straightedge constructions must not depend on measurement marks.
Drawing arcs that are too small is a problem because the arcs may not intersect clearly, so the needed construction point cannot be identified accurately.
Erasing construction arcs too early is wrong because those marks show why two distances or angles are congruent.
Assuming a diagram is accurate without checking construction steps is wrong because the proof comes from equal radii, intersections, and congruent angles, not from appearance.

Practice Questions

1 Construct a copy of $\overline{AB}$ with length $7\text{ cm}$ starting at point $P$ , and label the endpoint $Q$ so that $PQ = 7\text{ cm}$ .
2 Construct the perpendicular bisector of a segment $\overline{MN}$ where $MN = 8\text{ cm}$ , then state the length of each half of the segment.
3 Construct an angle bisector for an angle with measure $68^\circ$ , then find the measure of each smaller angle.
4 Explain why the perpendicular bisector construction works using the idea that points on the bisector are equidistant from the endpoints of the segment.

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