Practical geometry is the branch of mathematics that deals with constructing geometric figures accurately using a ruler, compass, protractor and set squares. Unlike theoretical geometry where figures are only studied, practical geometry requires careful drawing with exact measurements. In this chapter we learn to construct lines, angles, and triangles satisfying given conditions. Construction sharpens spatial reasoning and precision, skills that are essential in engineering, architecture, surveying and design.
The constructions in this chapter are based on the criteria for congruence of triangles. For example, if we know two sides and the included angle of a triangle, we can construct it using the SAS criterion. Similarly, SSS, ASA and RHS give methods for construction with other combinations of measurements. We also learn to construct parallel lines, perpendiculars and angle bisectors using a ruler and compass, which are the basic tools used in all advanced constructions.
All construction lines should be drawn lightly so that they can be erased after the figure is complete, while the required figure is drawn with darker, clear lines. Measurements should be taken accurately and the compass should be tightened to avoid the radius changing while drawing arcs.
To draw a line parallel to a given line l through a point P outside it: 1. Take a point Q on line l and join P to Q. 2. Using the protractor or compass, copy the angle made by QP with l at P so that the new angle is in the alternate position. 3. Draw the ray through P along the copied angle; this line is parallel to l.
The construction works because a pair of equal corresponding (or alternate interior) angles ensures the lines are parallel.
Given the lengths AB, BC and CA of a triangle: 1. Draw the longest side, say BC, with the ruler. 2. With the compass set to length AB, draw an arc from point B. 3. With the compass set to length CA, draw an arc from point C to intersect the previous arc. The intersection is vertex A. 4. Join A to B and A to C. The triangle ABC is complete.
This construction is possible only if the given sides satisfy the triangle inequality: the sum of any two sides must be greater than the third side.
Given sides AB, AC and the included angle A: 1. Draw one side, say AB, of the given length. 2. At A, draw the given angle using a protractor, and mark the direction of AC. 3. With the compass set to length AC, mark point C on the ray. 4. Join B to C. Triangle ABC is constructed.
The constructed triangle is unique because the SAS criterion guarantees a single triangle with these measurements.
Given angles A and B and the included side AB: 1. Draw side AB of the given length. 2. At A, construct angle A with a protractor, drawing a ray. 3. At B, construct angle B, drawing another ray. 4. The point where the two rays meet is vertex C. Triangle ABC is constructed.
The sum of the two given angles must be less than 180 degrees, since the third angle is 180 - (A + B).
Given the hypotenuse and one side of a right triangle: 1. Draw the given side, say BC, and at C construct a 90 degree angle. 2. On the perpendicular ray, mark the length of the side, giving point A. 3. Join A to B. The triangle ABC has the right angle at C, side BC given, and hypotenuse AB equal to the given length.
The RHS construction follows directly from the RHS congruence criterion for right triangles.
To construct the perpendicular bisector of segment AB: 1. With the compass open more than half of AB, draw arcs from A above and below AB. 2. Without changing the radius, draw similar arcs from B, intersecting the first arcs. 3. Join the two intersection points. This line is the perpendicular bisector of AB, and its point of crossing is the midpoint of AB.
To bisect an angle ABC: 1. With the compass at B, draw an arc cutting BA and BC at points D and E. 2. With the same radius, draw arcs from D and E intersecting at F. 3. Join B to F. The line BF bisects angle ABC into two equal angles.
| Criterion | Given Measurements | Steps in Brief |
|---|---|---|
| SSS | Three sides | Draw one side, arc from each end |
| SAS | Two sides and included angle | Draw side, make angle, mark other side |
| ASA | Two angles and included side | Draw side, make angles at both ends |
| RHS | Hypotenuse and one side | Draw side, right angle, join hypotenuse |
| Construction | Key Idea | Tools |
|---|---|---|
| Parallel line through a point | Copy angle in alternate position | Ruler, compass/protractor |
| Perpendicular bisector | Equal arcs from both ends | Ruler, compass |
| Angle bisector | Equal arcs from both arms | Ruler, compass |
| Right angle | 90 degree construction | Protractor/set square |
Practical geometry transforms the theorems of triangles and lines into careful, measurable drawings. Using a ruler, compass and protractor, we construct parallel lines, perpendicular bisectors, angle bisectors and triangles satisfying the SSS, SAS, ASA and RHS criteria. Each construction is rooted in a congruence criterion, connecting the abstract ideas of the earlier chapters with concrete craftsmanship. Checking validity conditions before construction, such as the triangle inequality, ensures we never attempt impossible figures. Mastery of these constructions builds the precision and spatial intuition needed for higher geometry, technical drawing and design.