Chapter 14 — Practical Geometry

Introduction
In Practical Geometry (Class 6), we use simple drawing and measuring instruments to make accurate figures. Each instrument has a clear purpose: measuring lengths, drawing straight lines, constructing angles and circles, or copying measurements.

Common Instruments and How to Use Them

• Ruler / Scale: A straight edge marked in centimetres (cm) and millimetres (mm). Use it to measure and draw straight lines. Align the 0 mark with one end of the segment and read at the other end. Keep the ruler flat and read at eye level.
• Protractor: A semicircular (180°) or full circular (360°) tool for measuring or drawing angles. To measure: place the protractor's centre (origin) at the vertex of the angle, align the baseline with one side, and read the degree where the other side crosses the scale.
• Compass (Pair of Compasses): Used to draw circles or arcs and to transfer equal lengths. Fix one point (the needle) at the centre, open the legs to the required radius using the scale on the compass or a ruler, then rotate the pencil leg around the centre keeping the needle steady.
• Divider: Similar to a compass but with two pointed ends. Used to transfer or compare lengths on a drawing (step-off equal segments along a line).
• Set Squares: Two triangular plastic/metal instruments: 45°-45°-90° and 30°-60°-90°. Use them to draw perpendiculars, and to quickly draw 30°, 60°, and 45° angles. Place one against a ruler or T‑square for stable drawing.
• Pencil, Eraser, Sharpener: Use a well-sharpened pencil (HB/2H for fine lines). Erase lightly to avoid damaging paper. Keep pencils sharp for accuracy.
• T‑square / Drawing board (optional for precise work): Helps draw parallel lines and align set squares when drawing technical figures.

Care and Safety
Keep instruments clean and dry. Close compasses when not in use. Carry instruments in a box or pouch to avoid bending or blunting. Use the sharp point of the compass/divider carefully to prevent injury.

Practical Tips — Typical Procedures

• To draw a line segment of given length: set the ruler, mark the endpoints at the correct measurement, then draw through the marks using the ruler's edge.
• To construct a circle with radius r: place compass needle at centre point, open to r using scale or ruler, draw circle by rotating pencil leg while keeping needle fixed.
• To measure an angle: place protractor centre on vertex, align baseline, read the inner or outer scale depending on orientation.
• To draw a perpendicular from a point on a line: use a set square against a ruler (or T‑square), place the point, and draw the perpendicular edge guideline.

Why accuracy matters
Accurate use of these instruments ensures correct constructions and measurements — an essential foundation for later geometry, technical drawing, engineering, and everyday tasks like carpentry or sewing.

Practical geometry starts with a set of basic terms and ideas used to describe and construct shapes. These simple building blocks — point, line, line segment, ray, angle, plane and circle — are the language of geometry. Knowing them and how to draw them with instruments (ruler, compass, protractor) helps in constructing and reasoning about figures.

• Point: A location in space with no length, width or thickness. It is denoted by a capital letter, e.g., A.
• Line: A straight one-dimensional figure extending infinitely in both directions. Represented with arrows at both ends, e.g., line l.
• Line segment: Part of a line with two endpoints, e.g., segment AB. Its length can be measured.
• Ray: A part of a line that starts at a point and extends infinitely in one direction, e.g., ray AX.
• Collinear points: Points that lie on the same straight line. Non-collinear do not lie on a single line.
• Plane: A flat two‑dimensional surface extending infinitely in all directions (like a sheet of paper).
• Angle: Formed by two rays (arms) with a common endpoint called the vertex. Written as ∠AOB where O is the vertex.
• Types of angles: right (90°), acute (<90°), obtuse (>90° and <180°), straight (180°), reflex (>180°). Also complementary (sum 90°) and supplementary (sum 180°) pairs.
• Parallel lines: Two lines in a plane that never meet (equal distance apart). Denoted by l ∥ m.
• Intersecting lines: Lines that meet at a point. If they meet at right angles they are perpendicular (denoted by ⟂).
• Polygon basics: A closed figure made of straight segments (triangle, quadrilateral). Vertex = corner where two segments meet.
• Circle basics: Set of points at a fixed distance (radius) from a centre. Key parts: centre (O), radius (r), diameter (d = 2r), chord, tangent.
• Instruments: Ruler (draw/measure straight lines), compass (draw arcs/circles, transfer distances), protractor (measure angles), set-square (draw perpendiculars/parallel lines).

Understanding these definitions and being able to sketch them (using labels, arrowheads, tick marks for equal lengths, small squares for right angles) is the first step in practical geometry.

What is a line segment? A line segment is the part of a straight line bounded by two distinct end points. If the end points are A and B, the segment is written as AB (or \overline{AB}). The length of segment AB is the distance between points A and B.

Basic ideas and properties

• Endpoints: Two points that determine the segment (A and B).
• Interior points: Points that lie between A and B on the same straight line.
• Notation: Segment AB is different from the line AB (which extends infinitely) and the ray AB (which extends in one direction).

Units and measurement

Lengths are measured using standard units: millimetre (mm), centimetre (cm), metre (m) etc. Use a ruler (cm/mm markings) for direct measurement or a divider for transferring lengths. Important conversions: 1 cm = 10 mm, 1 m = 100 cm.

How to measure a line segment (using a ruler)

1. Place the ruler so that the zero mark aligns exactly with one endpoint (A).
2. Read the length at the other endpoint (B). That reading is the length AB.
3. If the ruler’s zero is worn, use any clear mark (e.g., 1 cm) and subtract the offset: length = reading at B − reading at A.

How to draw a line segment of a given length (using a ruler)

1. Decide the required length (for example, 6.7 cm).
2. Place the ruler on paper and mark a point at 0 (or any reference mark).
3. From the reference, mark the required length on the paper (at 6.7 cm) and label the points A and B.
4. Join A and B with a straightedge (ruler edge) to get the segment AB.

Using a compass to transfer a length

1. To copy a segment AB to another location, place the compass point at A and the pencil at B, adjust the compass to that span.
2. Without changing the compass width, move the compass to the new location and mark an arc from the new starting point; join start and the marked point to get a congruent segment.

Accuracy tips

• Start reading at 0 on the ruler; if not possible, subtract the offset.
• Keep the ruler and paper steady; use a sharp pencil and a clear ruler edge.
• Use a divider when transferring a length to avoid parallax error.

Connections to number-line idea

If points A and B have coordinates x_A and x_B on a number line then length AB = |x_B − x_A|. This gives a clear visual way to understand subtraction as distance.

Perpendicular Lines: Two lines are said to be perpendicular if they meet (intersect) at a right angle (90°). If line l and line m meet at point O and ∠AOB = 90°, then l ⟂ m.

Perpendicular from a Point to a Line (construction using compass and ruler):

1. When the point P lies on the line AB: place the compass at P and draw arcs on both sides of P cutting AB at two points. With these two points as centers and same radius, draw two arcs (above and below AB). The line through their intersection and P is perpendicular to AB.
2. When the point P is not on the line AB: with center P and a radius large enough to cut AB at two points, draw an arc meeting AB at X and Y. With centers X and Y and same radius, draw two arcs that meet at Q (on the side of P). Line PQ is perpendicular to AB.

Perpendicular Bisector of a Line Segment: The perpendicular bisector of a segment AB is a line that is perpendicular to AB and passes through its midpoint M. It divides AB into two equal parts and makes a 90° angle with AB.

Construction of Perpendicular Bisector (compass & ruler):

1. With center A and radius more than half of AB draw an arc above and below the segment.
2. With the same radius and center B draw two arcs meeting the first pair at two points P and Q (one above and one below AB).
3. Draw line PQ. This line PQ is the perpendicular bisector of AB and meets AB at its midpoint M.

Key Properties:

• The perpendicular bisector passes through the midpoint of the segment.
• Any point on the perpendicular bisector is equidistant from the segment’s endpoints: if X lies on the perpendicular bisector of AB, then XA = XB.
• Perpendicular lines make a right angle (90°).
• In a triangle, the three perpendicular bisectors of the sides meet at one point called the circumcenter — the center of the circle that passes through all three vertices.

Why the construction works (simple idea): When you draw equal arcs from A and B that intersect at P and Q, triangles APA' and BPB' formed are congruent by RSR (same radii and common side), so PQ cuts AB at the midpoint and at right angle. Also any point on PQ is at equal distance from A and B because it lies on equal-radius arcs from A and B.

What is an angle?
An angle is formed by two rays (or line segments) that meet at a common end point called the vertex. The amount of turning from one ray to the other is the measure of the angle. The unit of angle measure commonly used in school is the degree (°).

Types of angles

• Acute angle: < 90°
• Right angle: = 90°
• Obtuse angle: > 90° and < 180°
• Straight angle: = 180°
• Reflex angle: > 180° and < 360°
• Full/Complete angle: = 360°

Measuring an angle (using a protractor)

1. Place the midpoint (center hole) of the protractor exactly at the vertex of the angle.
2. Align the zero line (baseline) of the protractor with one ray of the angle.
3. Read the number on the curved edge where the other ray meets the protractor. Use the inner or outer scale based on how you aligned the baseline.
4. If reading is unclear, mark the point where the ray meets the scale and measure from the vertex to that mark.

Constructing a given angle (with protractor)

1. Draw a ray OA (this will be one side of the angle).
2. Place protractor center at O and align baseline with OA.
3. On the protractor read the given degree measure and make a small mark on the paper at that degree.
4. Remove protractor and draw ray OB from O through the mark. ∠AOB is the required angle.

Copying an angle (using compass and ruler, without protractor)
This classical method reproduces the same angle size at a different location:

1. Let ∠XOY be the angle to copy. Draw a ray OA where you want the copy's vertex to be.
2. With center O (original angle), draw an arc that cuts both rays XO and YO at points A and B.
3. With the same compass width, draw an arc from the new vertex O' so it cuts OA at A'.
4. Measure the distance between A and B with the compass (the chord length on the first arc).
5. With the same compass setting, from A' draw an arc that intersects the arc from step 3 at point B'.
6. Join O' to B'. The angle A' O' B' is the copy of ∠XOY.

Bisecting an angle (using compass and ruler)

1. Given ∠AOB, place the compass at O (vertex) and draw an arc that cuts both OA and OB at P and Q respectively.
2. With the same radius, draw arcs center P and center Q so that they meet at R (choose radius more than half of PQ so arcs intersect).
3. Draw ray OR. OR is the angle bisector and divides ∠AOB into two equal angles.

Tips and common mistakes

• Always place the protractor correctly: centre at vertex and baseline along one ray.
• When copying without a protractor, use the same compass widths and do not change them between steps that must match.
• For bisecting, ensure arcs from P and Q intersect; if not, increase compass radius.

Why this matters (real-life uses)
Angles are everywhere: in clocks (angles between hands), cutting pieces (baking, carpentry), road junction design, architecture and art, navigation and robotics orientation. Knowing how to measure, construct, copy and bisect angles helps in precise drawing and planning.

Definition: Parallel lines are two lines in a plane that never meet, no matter how far they are extended. We write l || m to mean line l is parallel to line m.

Basic properties:

• If two lines are parallel, the distance between them is the same at every point.
• When a third line (a transversal) cuts two parallel lines, several angle relationships appear: corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (sum to 180°).

Angles formed by a transversal (short summary):

• Corresponding angles: lie in the same relative position at each intersection. If lines are parallel, corresponding angles are equal.
• Alternate interior angles: lie on opposite sides of the transversal and between the two lines. If lines are parallel, alternate interior angles are equal.
• Consecutive interior (co-interior) angles: lie on the same side of the transversal between the lines. If lines are parallel, these add up to 180°.

Constructions (Class 6 level):

1. Using a set-square (sliding method):
   1. Given line l and point P not on l, place the set-square so one of its edges lies on l.
   2. Slide the set-square keeping that edge on l until the opposite edge passes through P.
   3. Draw the line along the opposite edge through P — this line is parallel to l.
2. Using a ruler and compass (copy corresponding angle):
   1. Given line l and point P not on l, draw any line (transversal) through P that meets l at A.
   2. At point A on line l, identify the corresponding angle you want to copy (for example the angle between l and the transversal on the side where the new line should be).
   3. Using the compass, copy that angle at P (use the standard angle-copying steps: draw an arc cutting both sides of the angle at A, reproduce the same arc at P, measure the distance between the arc–intersection points and copy it; then join P to the copied point).
   4. The line through P so constructed will be parallel to l because corresponding angles are equal.

How to check your construction: After drawing the new line through P, draw a transversal and measure corresponding or alternate interior angles with a protractor — they should be equal. Alternatively, in coordinate or analytic view, equal slopes confirm parallelism.

Practical tips: Use a sharp pencil, steady compass width when copying angles, and a clear set-square slide for accurate results. Label points and angles while constructing to avoid confusion.

What is a triangle? A triangle is a three‑sided polygon made by joining three non‑collinear points (vertices) with three straight line segments (sides). Common tools for constructions are a ruler/scale, compass, and pencil.

Key ideas used in constructions

• Triangle inequality: For any triangle with side lengths a, b, c, each pair must satisfy a + b > c, b + c > a, and c + a > b. If this fails, a triangle cannot be constructed.
• Sum of interior angles: The three interior angles always add to 180° (useful to check angle constructions).
• Accuracy: Use the compass to copy lengths and draw arcs; use the ruler only for straight lines (do not measure angles with the ruler).

Standard construction cases (with step outlines)

1. SSS (three sides given: a, b, c)
   1. Draw the longest side (say c) as a straight line segment XY of length c.
   2. With center X and radius a, draw an arc. With center Y and radius b, draw another arc. The intersection gives vertex Z.
   3. Join Z to X and Y. Triangle XYZ has sides a, b, c.
2. SAS (two sides and included angle given: a, b and angle between them)
   1. Draw one side (say side of length b) as XY.
   2. At X, construct the given angle using compass and straightedge.
   3. Along the ray of that angle, mark point Z so that XZ = a (use compass). Join Z to Y.
3. ASA (two angles and included side given: ∠A, ∠B and side c between them)
   1. Draw the given side AB = c.
   2. At A, construct angle equal to given ∠A. At B, construct angle equal to given ∠B.
   3. The two rays meet at C. Join C to A and B to complete ΔABC.
4. RHS (right triangle: right angle, hypotenuse, one side) — for right triangles
   1. Draw the hypotenuse AB.
   2. At A, draw a line perpendicular to AB (construct 90°). On that line mark the given side length from A to C.
   3. Join C to B to form the right triangle.

Practical tips

• Always check the triangle inequality before attempting SSS constructions.
• When copying an angle, use the compass to transfer arc intersections for accuracy.
• Label each vertex and side clearly (A, B, C and a, b, c) to avoid confusion.

Why learn these? Triangle constructions teach precision, logical steps and fundamental geometric relationships used in engineering, drafting and everyday design.

What this topic covers
Practical Exercises and Applications in Class 6 Practical Geometry means learning how to make accurate drawings and constructions using basic instruments (ruler/scale, compass, protractor, pencil, set-square) and applying these constructions to everyday problems. Key skills include drawing/measuring line segments and angles, bisecting segments and angles, constructing perpendiculars, and making simple scale drawings and triangle constructions.

Basic steps and conventions

• Always label points (A, B, C...), arcs and intersection points.
• Use a sharp pencil and firm compass settings; draw light construction arcs and darken final lines.
• Mark the scale and a north arrow for maps or real-life layouts.

Typical constructions (with short descriptions)

• Perpendicular bisector of a segment AB: With A and B as centers and radius > half AB, draw two arcs above and below the line. Join the arc intersections — the line through them is the perpendicular bisector and meets AB at its midpoint.
• Angle bisector of ∠ABC: From B draw an arc that meets BA and BC at two points, then with these two points as centers draw two arcs that intersect. Join B to the intersection — that ray bisects the angle.
• Perpendicular from a point P to a line l: If P is on l, use compass to create a right angle (or use a set-square). If P is not on l, draw equal arcs from P to meet l at two points, then construct their perpendicular bisector; it will pass through P and meet l at a right angle.
• Triangle constructions: Using given data (e.g., three sides SSS or two sides and included angle SAS), draw a base, use compass arcs to locate the third vertex and join to complete the triangle.
• Scale drawings & maps: Convert actual lengths to drawing lengths using a chosen scale (for example 1:500 means 1 cm on paper = 500 cm in reality). Draw using the scale, add legend/scale bar and north arrow.

Applications
These constructions are used in architecture (floor plans), carpentry (right angles, bisecting), engineering sketches, map-making (scale plans of parks, playgrounds), school laboratory setups (positioning instruments), and art/design (accurate geometric shapes).

Tips for students

• Practice each construction several times; label each step and justify intersections.
• Check accuracy: measure resulting angles and lengths with protractor/ruler.
• When making scale drawings, include a clear scale bar (for example, a 5 cm bar that represents 25 m).