MATHEMATICS CLASS NOTES: ANGLES AND POLYGONS

Level: Junior Secondary School (JSS 2) Subject: Mathematics Topic: Angles and Polygons Duration: 2 Weeks Curriculum: Nigerian National Curriculum (NERDC)

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1. COMPREHENSIVE CORE CONCEPTS

A. Understanding Angles: The Foundation of Geometry

In the simplest terms, an angle is a measure of "turn" or the opening between two straight lines that meet at a common point.

• The Vertex: This is the exact point where the two lines meet.
• The Arms: These are the two rays or line segments that form the angle.
• The Unit: We measure angles in degrees (°). A full circle is $360^\circ$, which represents a complete rotation.

Why do we study angles? Angles are not just marks on paper; they determine how stable a building is, how a satellite orbits the earth, and how a tailor cuts fabric to fit the human body. When you open a door, you are creating an angle between the door and the wall. If you open it halfway to a perpendicular position, you have created a Right Angle ($90^\circ$).

B. Geometrical Construction of Special Angles

In Mathematics, "Construction" is different from "Drawing." We do not use a protractor to draw the angle; instead, we use a pair of compasses and a straightedge (ruler) to build the angle from scratch. This method ensures absolute geometric precision.

1. Constructing $120^\circ$ (The Double $60^\circ$)

Since we know that the first arc from a point on a line (using the same radius) gives us $60^\circ$, we simply repeat the process.

• Step 1: Draw a straight line $AB$ and mark a point $O$ as your vertex.
• Step 2: Open your compass to a convenient width. Place the point at $O$ and draw a wide arc (a semi-circle) that cuts line $AB$ at point $P$.
• Step 3: Without changing the compass width, place the point at $P$ and draw an arc that cuts the first arc at $Q$. (This is the $60^\circ$ mark).
• Step 4: Place the compass point at $Q$ and draw another arc further along the semi-circle to get point $R$.
• Step 5: Draw a line from $O$ through $R$. The angle $\angle ROB$ is $120^\circ$.

2. Constructing $75^\circ$ (The Midpoint of $60^\circ$ and $90^\circ$)

$75^\circ$ is exactly halfway between $60^\circ$ and $90^\circ$. To get it, you must first have both a $60^\circ$ and a $90^\circ$ construction on the same point.

• Step 1: Construct a $90^\circ$ angle (perpendicular bisector method) and a $60^\circ$ arc on the same base line.
• Step 2: Focus on the arc space between the $60^\circ$ mark and the $90^\circ$ mark. This space represents $30^\circ$ ($90 - 60 = 30$).
• Step 3: Use your compass to bisect this $30^\circ$ segment. Place the compass at the $60^\circ$ mark and draw an arc; then place it at the $90^\circ$ mark and draw another arc to intersect the first.
• Step 4: Draw a line from the vertex through this intersection. $60^\circ + 15^\circ = 75^\circ$.

3. Constructing $105^\circ$ and $135^\circ$

• $105^\circ$: This lies halfway between $90^\circ$ and $120^\circ$. Construct both $90^\circ$ and $120^\circ$, then bisect the $30^\circ$ gap between them ($90 + 15 = 105$).
• $135^\circ$: This lies halfway between $90^\circ$ and $180^\circ$. Construct a $90^\circ$ angle on a straight line. Bisect the $90^\circ$ angle on the left side of the perpendicular. $90^\circ + 45^\circ = 135^\circ$.

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C. Polygons: Definition and Classification

A Polygon is a flat, closed shape with at least three straight sides.

• Regular Polygons: All sides are the same length, and all interior angles are equal (e.g., an Equilateral Triangle).
• Irregular Polygons: These have sides and angles of different sizes.

Common Polygons to Remember:

1. Triangle: 3 sides
2. Quadrilateral: 4 sides (Square, Rectangle, Trapezium)
3. Pentagon: 5 sides
4. Hexagon: 6 sides
5. Heptagon: 7 sides
6. Octagon: 8 sides

D. Sum of Interior Angles Formula

The total sum of all angles inside a polygon increases as the number of sides ($n$) increases. We use a specific formula to find this sum: $\text{Sum} = (2n - 4) \text{ right angles}$ Since one right angle = $90^\circ$, the formula is: $\text{Sum} = (2n - 4) \times 90^\circ$

Why does this work? Any polygon can be split into $(n - 2)$ triangles. Since every triangle has $180^\circ$, the sum is $(n - 2) \times 180^\circ$. If you expand $(n - 2) \times 180$, it is the same as $(n - 2) \times 2 \times 90$, which gives us $(2n - 4) \times 90$.

Example: Find the sum of interior angles of an Octagon ($n = 8$).

1. $\text{Sum} = (2 \times 8 - 4) \times 90^\circ$
2. $\text{Sum} = (16 - 4) \times 90^\circ$
3. $\text{Sum} = 12 \times 90^\circ = 1080^\circ$.

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2. REAL-WORLD EXAMPLES

1. Civil Engineering (The Third Mainland Bridge): Engineers use angles to distribute the weight of the bridge. The triangular supports (trusses) seen on bridges are polygons chosen because they are the most stable shape under pressure.
2. Traditional Fashion (Agbada Design): When a tailor cuts the neckline of an Agbada or the "U" shape of a Buba, they are essentially working with polygons and angles. A perfectly balanced sleeve requires a specific angle of attachment to the main body.
3. Solar Energy: In Nigeria, to maximize the efficiency of solar panels, they must be tilted at an angle (usually between $15^\circ$ and $25^\circ$ depending on the state) to catch the maximum rays of the sun.
4. Sports (Football): A striker like Victor Osimhen knows that the "angle of approach" to the ball determines where it lands in the net. Shooting from a "tight angle" means the polygon formed by the player and the goal posts is very narrow.

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3. PROJECT-BASED LEARNING: "The Geometric Architect"

Objective: To design a "Smart Kiosk" floor plan using precise angle construction and polygon calculations.

Materials Needed:

• A4 Plain Paper
• Mathematical Set (Ruler, Compass, Pencil)
• Colored Pens

Step-by-Step Instructions:

1. The Foundation: Use your ruler to draw a base line. Construct a Regular Hexagon (6 sides) with each side measuring 5cm. This will be the main floor of your kiosk.
2. The Entrance: At the front vertex of your hexagon, construct an angle of $105^\circ$ to represent the opening of a decorative glass door.
3. The Counter: Inside the kiosk, construct a quadrilateral (a rectangle) that takes up one side.
4. The Calculation: On the side of your drawing, show the mathematical proof for the hexagon's angles. Use the formula: $(2 \times 6 - 4) \times 90^\circ$.
5. The Presentation: Label each interior angle and each side length clearly.

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4. HOME PRACTICE ACTIVITIES

Activity 1: The Shadow Angle Experiment

• Materials: A stick (30cm), a ruler, and a sunny day.
• Procedure: At 12:00 PM and 4:00 PM, place the stick upright in the ground. Measure the length of the shadow. The stick and the shadow form a $90^\circ$ angle at the base.
• Outcome: Draw the triangle formed by the stick and the shadow on paper. Use your protractor to measure the angle at the tip of the shadow. You will see how the angle of the sun changes throughout the day!

Activity 2: The Polygon Scavenger Hunt

• Task: Find one object in your house for each of the following: a triangle, a quadrilateral, a pentagon, and a hexagon. (Hints: Hangers, Floor tiles, Nut-bolts, Wall clocks).
• Task: For the Hexagon or Octagon object, use the formula $(2n - 4) \times 90^\circ$ to calculate what the sum of its interior angles should be.

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5. LIFE SKILLS INTEGRATION

• Precision and Accuracy: In construction, being off by just $2^\circ$ can cause a roof to leak or a wall to crack. Learning to construct angles accurately builds a habit of precision that is vital for careers in Medicine (Surgery), Aviation, and Engineering.
• Spatial Intelligence: Understanding how polygons fit together (tessellation) is a skill used by Interior Designers and Warehouse Managers to fit the maximum amount of furniture or goods into a limited space.
• Critical Thinking: When you use the formula $(2n - 4) \times 90^\circ$, you aren't just calculating; you are using logic to understand the internal structure of shapes.

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6. ASSESSMENT THROUGH APPLICATION

1. The "Unknown Angle" Challenge: You are given a regular pentagon. If the sum of the angles is $(2 \times 5 - 4) \times 90 = 540^\circ$, what is the size of each individual interior angle? (Answer: $540 \div 5 = 108^\circ$).
2. The Practical Construction: On a blank sheet, construct a $135^\circ$ angle. Now, imagine this is a ramp for a wheelchair. Bisect it to create two equal $67.5^\circ$ angles.
3. The Error Finder: A student calculates the sum of angles of a Quadrilateral as $540^\circ$. Use the $(2n - 4)$ formula to prove why they are wrong and show the correct calculation.

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7. STUDENT REFLECTION QUESTIONS

• Why do we use a compass and ruler instead of just a protractor for construction? (Hint: Think about precision).
• If you add more sides to a polygon, what happens to the total sum of the interior angles?
• Can you find any regular polygons in nature? Why do you think bees use hexagons for their honeycombs instead of circles?
• How does knowing the sum of angles help a carpenter who is building a wooden roof frame?