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USA - Geometry Geometric constructions

Basic constructions (segment, angle, bisectors, perpendiculars, parallels)

20 practice questions 2 video lessons Theory + worked examples

Basic Geometric Constructions

Texas Geometry (TEKS) • Standard G.5(B) • Geometric Constructions

Basic Geometric Constructions is the opening topic of Geometric Constructions in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.5(B), which requires students to construct congruent segments and angles, angle and segment bisectors, perpendiculars, and parallels.

The basic constructions use only a compass and straightedge to copy segments and angles and to build bisectors, perpendiculars, and parallels.

Texas Geometry (TEKS) › Geometric Constructions › Basic Geometric Constructions  —  Standard G.5(B)

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Practice questions

Every question with a fully worked solution.

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  • Constructions - GCSE Maths Watch
  • Geometry Constructions (15 Must Know Types) with Compass and Straightedge Watch
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Theory

A construction creates a precise figure using only a compass and straightedge — no rulers or protractors. The core constructions are:

  • Copy a segment or an angle.
  • Perpendicular bisector of a segment (equal arcs from each endpoint).
  • Angle bisector (arcs from the sides meeting inside).
  • Perpendicular to a line through a point, and a parallel line through a point.
Constructions work because of congruence: equal compass radii create congruent triangles, which force the exact relationship.
Constructing a perpendicular bisector Equal arcs from each endpoint meet at two points; the line through them is the perpendicular bisector. A B arcs from A and B meet on the bisector
Perpendicular bisector: equal arcs from \(A\) and \(B\) meet on it.
Constructing an angle bisector An arc across both rays, then equal arcs meeting inside, locate the ray that bisects the angle. bisector splits the angle in two
Angle bisector: arcs locate the dividing ray.

The compass rule behind bisectors:

\[\text{equal radii}\Rightarrow \text{equidistant points}\Rightarrow \text{bisector}\]
equal compass radii create equidistant points that lie on the bisector
Radius \(>\dfrac12 AB\) guarantees the perpendicular-bisector arcs intersect.

How to construct a perpendicular bisector

  1. Open the compass wider than half the segment.
  2. Draw arcs from each endpoint, above and below.
  3. Mark the two intersection points.
  4. Draw the line through them — it is perpendicular and bisects the segment.
Example 1 — The tools
What two tools are used in classical geometric constructions?
Solution

A compass (for arcs and equal lengths) and a straightedge (for lines) — no measuring.

a compass and a straightedge
Example 2 — Perpendicular bisector
To construct the perpendicular bisector of \(\overline{AB}\), what radius should the compass arcs use?
Solution

Any radius greater than half of \(AB\), so the arcs from \(A\) and \(B\) intersect on both sides.

\(\text{radius}\)\(>\)\(\dfrac{1}{2}AB\)
a radius greater than half of AB
Example 3 — Angle bisector
What does the angle-bisector construction produce?
Solution

A ray from the vertex that divides the angle into two equal angles.

a ray dividing the angle into two equal angles
Example 4 — Parallel line
How is a line parallel to a given line constructed through a point?
Solution

Copy one of the angles the given line makes with a transversal at the point; equal corresponding angles force the new line to be parallel.

copy a corresponding angle to make the lines parallel

Common pitfalls

No measuring. Constructions use only compass and straightedge, not a ruler or protractor.
Keep the compass width fixed while making the matching arcs.
Perpendicular-bisector arcs need radius \(>\dfrac12 AB\), or they won't meet.

Frequently asked questions

What tools are used in a geometric construction?

A compass and a straightedge only — no ruler markings or protractor.

How do you construct a perpendicular bisector?

Draw equal arcs from each endpoint (radius more than half the segment); the line through their two intersections is the perpendicular bisector.

How do you construct an angle bisector?

Draw an arc across both sides of the angle, then equal arcs from those points; the ray from the vertex through their intersection bisects the angle.

Why do constructions work?

Because equal compass radii create congruent triangles, which force the exact geometric relationship being constructed.

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