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West Bengal Board
Class 7
Mathematics (গণিত প্রভা)
Ch 7: Construction of Specific Angles with Compass

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WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 7

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Construction of Specific Angles with Compass

Welcome to Chapter 7 "Construction of Specific Angles with Compass" of the West Bengal Board (WBBSE) Class 7 Mathematics curriculum. Built upon the TargetExams Gold-Standard 5-Step Pedagogy System, this module covers classical Euclidean geometry using only a straightedge (scale) and pair of compasses. Master 60° and 120° arc principles, angle bisection for 30°, 15°, 75°, 90° perpendicular construction, derived angles like 45°, 105°, 135°, angle copying, and the constructibility condition for multiples of 15°.

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Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

📐 Constructing Angles Without a Protractor: The Ancient Euclidean Secret

How did ancient architects construct laser-accurate right angles and hexagons without modern protractors?

In ancient Greece, Euclid established that the purest geometric truths require only two tools: an unmarked straightedge to connect two points, and a compass to draw circles. Protractors with degree markings did not exist!

Stepping off a compass radius around a circular perimeter divides it precisely into six equal 60° sectors, forming a regular hexagon. Bisecting these arcs unlocks 30°, 15°, 45°, 75°, and 90°. From astronomical sundials and Gothic cathedral arches to modern satellite dish designs, compass-and-straightedge geometry remains the pinnacle of mathematical precision.

Why This Chapter Matters

Before You Begin (Prerequisites)

Concepts of points, lines, rays, line segments, and circular arcs.
Handling a compass securely without letting the hinge radius slip.
Angle classification: acute (< 90°), right (= 90°), obtuse (> 90°), straight (= 180°).

What You Will Learn (Core Objectives)

Construct accurate 60° and 120° angles using the equilateral triangle arc principle.
Bisect any given angle precisely using intersecting equidistant circular arcs.
Construct a 90° right angle and the perpendicular bisector of a line segment.
Synthesize derived angles: 30°, 45°, 75°, 105°, 135°, and 150° using bisection and angle addition.
Articulate the constructibility criterion (angles must be multiples of 15°) and identify non-constructible angles.

Chapter Roadmap & Progression

1 1. Geometric Principle & Constructi...

2 2. Angle Bisection: Constructing 30...

3 3. The 90° Right Angle & Perpendicu...

4 4. Constructing 45°, 105°, and 135°...

5 5. Constructible Angles & Copying a...

Complete Concept Guide (100% Curriculum Coverage)

1. Geometric Principle & Construction of 60° and 120° Angles

1. The Intuition

An equilateral triangle has three identical side lengths and three equal angles of 60°. When you draw an arc from origin $O$ and cut it with the exact same compass radius, you are essentially creating two vertices of an equilateral triangle. Therefore, the angle subtended at the center is always exactly 60°.

2. Formal Concept & Construction Steps

Steps to Construct a 60° Angle:

Draw a base ray $OA$.
With center $O$ and any convenient radius, draw a circular arc cutting ray $OA$ at point $P$.
Without altering the compass radius, place the compass point at $P$ and cut the initial arc at point $Q$.
Draw ray $OQ$. The angle $\angle AOQ = \mathbf{60^\circ}$.

Constructing 120°: Keeping the radius unchanged, place the compass at $Q$ and cut the arc again at $R$. Ray $OR$ forms $\angle AOR = 60^\circ + 60^\circ = \mathbf{120^\circ}$.

3. Concrete Worked Example

Problem: Construct an angle of 60° at vertex $B$ on a ray $BC = 6\text{ cm}$ using compass and ruler only.

1) Draw ray $BC = 6\text{ cm}$.

2) With center $B$, draw a wide arc cutting $BC$ at point $X$.

3) With center $X$ and unchanged radius, draw an arc intersecting at $Y$.

4) Connect $B$ and $Y$, extending to $A$.

Verification: Measuring with a protractor confirms $\angle ABC = \mathbf{60^\circ}$.

4. Pitfall & Examiner Trap

⚠️ Critical Error: Slipping Compass Radius
If the compass joint loosens between drawing the main arc and cutting at $P$, the radius changes and the angle will be incorrect (e.g. 58° or 63°). Tighten the hinge screw before starting!

5. Why This Matters in Life

Tessellating hexagonal floor tiles and carbon graphene lattice structures are built entirely on 60° and 120° bonds.

2. Angle Bisection: Constructing 30°, 15°, and 75°

1. The Intuition

Bisecting an angle splits it into two equal halves. Bisecting 60° yields 30°; bisecting 30° yields 15°. Moreover, bisecting the 30° difference between 60° and 90° yields $60^\circ + 15^\circ = 75^\circ$!

2. Formal Concept & Bisection Pipeline

$\mathbf{30^\circ = \frac{60^\circ}{2}}$: Bisect the 60° angle.
$\mathbf{15^\circ = \frac{30^\circ}{2}}$: Bisect the 30° angle.
$\mathbf{75^\circ = 60^\circ + \frac{90^\circ - 60^\circ}{2} = 60^\circ + 15^\circ}$: Bisect the arc interval between 60° and 90°.

Bisection Procedure: From the two arc intersection points on the arms, draw intersecting arcs with a radius greater than half the distance between them. Draw a ray from the vertex through the intersection point.

3. Concrete Worked Example

Problem: Construct an angle of 75° with compass and ruler.

1) Construct 60° ray ($OQ$) and 90° ray ($OS$).

2) Identify arc intersection points $Q$ (at 60°) and $S$ (at 90°) on the main circular arc.

3) From $Q$ and $S$, draw arcs of equal radius crossing at point $T$.

4) Draw ray $OT$. Then $\angle AOT = 60^\circ + 15^\circ = \mathbf{75^\circ}$.

Answer: $\angle AOT = \mathbf{75^\circ}$

4. Pitfall & Examiner Trap

⚠️ Trap: Bisecting the Wrong Sector for 75°
Students often bisect between 0° and 60° (giving 30°) instead of bisecting between 60° and 90°. Ensure you place compass points on the 60° and 90° arc markers.

5. Why This Matters in Life

Solar panel mounting brackets in northern latitudes are angled at 30° to 45° to optimize photon absorption.

3. The 90° Right Angle & Perpendicular Bisector Construction

1. The Intuition

What lies exactly halfway between 60° and 120°? The arithmetic mean $\frac{60^\circ + 120^\circ}{2} = 90^\circ$! Bisecting the 60° gap between 60° and 120° naturally constructs a vertical, perpendicular right angle.

2. Formal Concept & Steps

Constructing 90° at a point on a ray:

Draw main arc cutting base ray at $P$. Cut 60° at $Q$ and 120° at $R$.
From $Q$ and $R$, draw intersecting arcs of equal radius above the line crossing at point $S$.
Draw ray $OS$. Then $OS \perp OA$ and $\angle AOS = \mathbf{90^\circ}$.

Perpendicular Bisector of Segment $AB$: With radius $> \frac{1}{2}AB$, draw arcs above and below from both $A$ and $B$. Connecting the intersections yields a line that cuts $AB$ into two equal halves at right angles.

3. Concrete Worked Example

Problem: Draw a line segment $AB = 7\text{ cm}$ and construct its perpendicular bisector.

1) Draw segment $AB = 7\text{ cm}$.

2) Open compass to $4.5\text{ cm}$ ($> 3.5\text{ cm}$). From $A$, draw arcs above and below.

3) From $B$ with same radius, intersect both arcs at $P$ and $Q$.

4) Join $P$ and $Q$. The line crosses $AB$ at midpoint $M$.

Result: $AM = MB = \mathbf{3.5\text{ cm}}$ and $\angle PMA = \mathbf{90^\circ}$.

4. Pitfall & Examiner Trap

⚠️ Fatal Mistake: Compass Width Less Than Half the Segment
If the radius is less than $\frac{1}{2}AB$, the two arcs will never cross! The radius must be strictly greater than half the segment length.

5. Why This Matters in Life

Civil engineering foundations and carpenter plumb bobs rely on perpendiculars to ensure vertical wall stability.

4. Constructing 45°, 105°, and 135° Angles

1. The Intuition

Bisecting a right angle (90°) yields 45°. Combining 90° with a bisected 90° on the other side yields $90^\circ + 45^\circ = 135^\circ$. Bisecting between 90° and 120° yields $90^\circ + 15^\circ = 105^\circ$.

2. Formal Concept & Relationships

Angle	Formulation	Construction Strategy
$\mathbf{45^\circ}$	$90^\circ / 2$	Bisect 90° right angle
$\mathbf{105^\circ}$	$90^\circ + 15^\circ$	Bisect 30° arc between 90° and 120°
$\mathbf{135^\circ}$	$90^\circ + 45^\circ$	Bisect 90° sector between 90° and 180° straight angle
$\mathbf{150^\circ}$	$120^\circ + 30^\circ$	Bisect 60° sector between 120° and 180°

3. Concrete Worked Example

Problem: Construct an angle of 135° with straightedge and compass.

1) Draw a straight line $AOB$ (straight angle $180^\circ$).

2) Construct perpendicular $OC \perp AB$ at $O$ ($\angle AOC = 90^\circ$ and $\angle BOC = 90^\circ$).

3) Bisect the left-hand 90° angle ($\angle AOC$) with ray $OD$.

4) Then $\angle BOD = \angle BOC + \angle COD = 90^\circ + 45^\circ = \mathbf{135^\circ}$.

Result: $\angle BOD = \mathbf{135^\circ}$

4. Pitfall & Examiner Trap

⚠️ Examiner Trap: Identifying 45° instead of 135°
In 135° constructions, students often accidentally shade the acute 45° supplementary wedge instead of the obtuse 135° angle.

5. Why This Matters in Life

Chamfered edges in mechanical drafting and stealth aircraft radar-deflecting fuselage panels use 45° and 135° facets.

5. Constructible Angles & Copying a Given Angle

1. The Intuition

Can any arbitrary angle be drawn using only a straightedge and compass? No! Classical geometry proves that only angles that are multiples of 15° (or their repeated bisections) can be constructed without a protractor.

2. Formal Concept & The 15° Divisibility Rule

The 15° Constructibility Axiom:
An integer angle $\theta$ is constructible with straightedge and compass if and only if it is a multiple of $15^\circ$:
$$\mathbf{\theta = 15^\circ \times k \quad (k \in \mathbb{Z}^+)}$$ Examples: $15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ, 90^\circ, 105^\circ, 120^\circ, 135^\circ, 150^\circ, 165^\circ$.

Non-constructible Angles: Angles such as $20^\circ, 40^\circ, 50^\circ, 70^\circ, 80^\circ$ cannot be constructed using compass and straightedge alone (they require a protractor).

3. Concrete Worked Example

Problem: Determine which of the following angles can be constructed using only ruler and compass: a) 40°, b) 105°, c) 67.5°

a) $40^\circ$: No, because 40 is not divisible by 15.

b) $105^\circ$: Yes, because $105 = 15 \times 7$.

c) $67.5^\circ$: Yes, because it is the bisection of 135° ($135^\circ / 2 = 67.5^\circ$).

4. Pitfall & Examiner Trap

⚠️ Exam Penalty: Erasing Construction Arcs
Never erase compass arcs! Examiners inspect intersecting arc marks to award full marks. Erasing them can result in zero credit.

5. Why This Matters in Life

Nautical navigation compasses partition directions into 32 points, using repeated bisections (45°, 22.5°, 11.25°) to chart course trajectories.

Key Formulas, Identities & Theorems

60° Construction

$$\text{Radius}(OA) = \text{Radius}(PQ)$$

First arc strike with same radius

120° Construction

$$60^\circ + 60^\circ = 120^\circ$$

Second arc strike with same radius

90° Right Angle

$$60^\circ + 30^\circ = 90^\circ$$

Bisect arc interval between 60° and 120°

45° Bisection

$$90^\circ / 2 = 45^\circ$$

Bisect 90° right angle

75° Angle

$$60^\circ + 15^\circ = 75^\circ$$

Bisect between 60° and 90°

135° Angle

$$90^\circ + 45^\circ = 135^\circ$$

Bisect between 90° and 180°

Conceptual Solved Examples & Case Studies

Example 1

State the step-by-step construction of a 45° angle with ruler and compass.

Step-by-Step Solution:

1) Draw base ray $OA$.
2) Construct a 90° angle at $O$ ($\angle AOB = 90^\circ$).
3) Let the circular arc cut $OA$ at $P$ and $OB$ at $Q$.
4) With radius $> \frac{1}{2}PQ$, draw intersecting arcs from $P$ and $Q$ meeting at $R$.
5) Ray $OR$ bisects $\angle AOB$ to give $\angle AOR = \mathbf{45^\circ}$.

Common Misconceptions & Examiner Traps

Common Misconception

Using a protractor when the question explicitly specifies compass only.

Scientific Reality & Correction

Examiners award zero marks if construction arcs are missing. Always use a compass.

Common Misconception

Opening compass less than half the distance during bisection.

Scientific Reality & Correction

Arcs will fail to cross. Always open compass $> 1/2$ of arc span.

Visual Learning & Conceptual Map

WBBSE Mathematics Foundations: The Compass Angle Navigator

Arc radius invariance, bisection pathways, and classical constructibility

1. Base 60°

1st Arc Cut

Unchanged radius

2. Right Angle 90°

Between 60° & 120°

Perpendicular intersection

3. Bisection Pipeline

30°, 45°, 75°

Radius > 1/2 arc distance

4. Constructibility Rule

Multiples of 15°

20°, 40°, 50° impossible

Chapter Summary & 10 Key Takeaways

Takeaway 1

60° Angle: Constructed by cutting a circular arc with its own radius.

Takeaway 2

120° Angle: Constructed by taking two consecutive 60° arc steps.

Takeaway 3

90° Angle: Formed by bisecting the arc between 60° and 120°.

Takeaway 4

Angle Bisection: Splits any angle into two equal halves.

Takeaway 5

Derived Angles: 30° (half of 60°), 45° (half of 90°), 75° (60° + 15°), 105° (90° + 15°).

Takeaway 6

Constructibility Condition: Only multiples of 15° can be constructed with straightedge and compass.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

Can an angle of 50° be constructed using only a ruler and compass?

Reveal Answer & Explanation

Answer: No. Because 50° is not a multiple of 15°. It requires a protractor.
Check if 50 is divisible by 15.

What two angle rays are bisected to construct a 75° angle?

Reveal Answer & Explanation

Answer: The 60° ray and the 90° ray ($60^\circ + 15^\circ = 75^\circ$).
75 lies halfway between 60 and 90.

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