How do you carry out the classic compass-and-straightedge constructions, and why do they work?
Perform compass-and-straightedge constructions: copy a segment and an angle; bisect a segment (perpendicular bisector) and an angle; construct a perpendicular and a parallel line; construct an equilateral triangle; and explain why each construction produces the intended figure.
A NY Regents Geometry answer on compass-and-straightedge constructions: copying segments and angles, perpendicular and angle bisectors, perpendicular and parallel lines, the equilateral triangle, and why each one works.
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What this topic is asking
The Regents Geometry exam (the Congruence, G-CO, cluster) wants you to carry out the classic compass-and-straightedge constructions and, often, to explain why they work. The core set is: copy a segment and an angle, bisect a segment (the perpendicular bisector) and an angle, drop or raise a perpendicular, construct a parallel line, and build an equilateral triangle. Constructions appear in Part I as recognition questions and in Parts II to IV as draw-and-justify tasks.
The toolkit and the rules
A construction is an exact geometric procedure using only two tools: a compass to copy distances (draw arcs of a set radius) and a straightedge to draw straight lines. You may not measure with a ruler or protractor, because the point is to produce the figure from geometric principles. The Regents expects all construction marks (arcs) to be left visible, since they are the evidence the construction was done correctly.
The perpendicular bisector and the angle bisector
Two constructions anchor almost all the others.
Both work for the same reason: the equal compass radii create congruent triangles, so equal distances or equal angles follow.
Building the others
The remaining constructions reuse these ideas. A perpendicular through a point on a line uses arcs to mark two equidistant points, then the perpendicular bisector of those points. A parallel line through a point is built by copying an angle (so corresponding angles are equal, forcing the lines parallel). An equilateral triangle on a segment comes from two arcs of radius centered at and ; their intersection gives , so the triangle is equilateral.
A clarifying point worth stressing is that the justification is almost always congruent triangles. Whenever a construction asks "explain why it works", the equal compass radii are equal sides, the constructed figure splits into two triangles, SSS makes them congruent, and CPCTC delivers the equal angle or equal distance. Reaching for that template answers nearly every construction justification on the exam.
Try this
Q1. Which two tools are allowed in a construction? [1 credit]
- Cue. A compass and a straightedge only (no ruler or protractor).
Q2. What single construction gives you a 60-degree angle directly? [1 credit]
- Cue. Constructing an equilateral triangle: each of its angles is 60 degrees.
Exam-style practice questions
Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Regents (style)2 marksPart I (multiple choice). A construction is performed by drawing two arcs of equal radius centered at the two endpoints of a segment, on both sides, then drawing the line through the two arc intersections. This construction produces (1) an angle bisector (2) the perpendicular bisector of the segment (3) a parallel line (4) a copy of the segment.Show worked answer →
The correct answer is (2).
Drawing equal-radius arcs from both endpoints creates two points each equidistant from the endpoints. Every point equidistant from the two endpoints lies on the perpendicular bisector, so the line through the two intersection points is the perpendicular bisector of the segment. The arcs must have radius greater than half the segment so they intersect.
Regents (style)2 marksPart II (constructed response). Using a compass and straightedge, construct the bisector of a given angle. Describe the steps and explain why the resulting ray bisects the angle. (A clear description with justification is required.)Show worked answer →
A 2-credit question: 1 credit for a correct construction description, 1 for the justification.
Place the compass at the vertex and draw an arc crossing both sides of the angle at points and . Without changing the radius (or with any fixed radius), draw arcs from and from that intersect at a point inside the angle. Draw the ray from the vertex through ; this ray bisects the angle. It works because and are equidistant from the vertex and is equidistant from and , making the two triangles formed congruent by SSS, so the two angles at the vertex are congruent by CPCTC. A description with no justification earns 1 credit.
Related dot points
- Represent and perform reflections, rotations, and translations using rules and the coordinate plane; recognize that rigid motions preserve distance and angle; and define congruence of two figures as the existence of a sequence of rigid motions mapping one onto the other.
A NY Regents Geometry answer on rigid motions: performing reflections, rotations, and translations on the coordinate plane, why they preserve distance and angle, and how a sequence of rigid motions defines congruence.
- Use the triangle congruence criteria (SSS, SAS, ASA, AAS, HL) to prove two triangles congruent, and use CPCTC (corresponding parts of congruent triangles are congruent) to justify further equal sides or angles.
A NY Regents Geometry answer on triangle congruence: the SSS, SAS, ASA, AAS, and HL criteria, why SSA and AAA fail, and using CPCTC to conclude further equal parts once triangles are proven congruent.
- Prove theorems about lines and angles (vertical angles, the angle relationships from parallel lines cut by a transversal) and about triangles (the angle sum is 180 degrees, the exterior angle theorem, the isosceles triangle base angles, and the midsegment theorem).
A NY Regents Geometry answer on proving angle and triangle theorems: vertical angles, parallel-line angle pairs, the triangle angle sum, the exterior angle theorem, isosceles base angles, and the midsegment theorem.
- Apply central angle, inscribed angle, chord, tangent, and secant relationships in a circle; compute arc length and sector area; and write and use the equation of a circle in the coordinate plane.
A NY Regents Geometry answer on circles: central and inscribed angles, the chord, tangent, and secant relationships, arc length and sector area, and the standard equation of a circle on the coordinate plane.
- Use the distance, midpoint, and slope formulas to analyze figures; determine parallel and perpendicular lines from slope; partition a directed segment into a given ratio; and write equations of lines satisfying given conditions.
A NY Regents Geometry answer on coordinate geometry: the distance, midpoint, and slope formulas, parallel and perpendicular slope conditions, partitioning a segment in a ratio, and writing equations of lines.
Sources & how we know this
- Regents Examination in Geometry — NYSED (2024)
- New York State Next Generation Mathematics Learning Standards — NYSED (2017)