Subject: Mathematics | Level: JHS (Basic 7 – 9) | Topic: Geometry – Angles, Triangles, Pythagoras | Difficulty: Intermediate
Introduction
Geometry is the branch of Mathematics that deals with shapes, sizes, positions, and the properties of space. It is one of the most practical areas of Mathematics, used in architecture, engineering, carpentry, design, and everyday measurements. At JHS level, Geometry covers angles, triangles, circles, polygons, and measurement. This lesson focuses on the key angle rules, triangle properties, and Pythagoras theorem that appear regularly in the BECE.
1. Types of Angles
- Acute angle: less than 90°
- Right angle: exactly 90°
- Obtuse angle: between 90° and 180°
- Straight angle: exactly 180°
- Reflex angle: between 180° and 360°
- Complete revolution: exactly 360°
2. Angle Rules (Must Know for BECE)
Angles on a Straight Line
All angles on a straight line add up to 180°.
Example: Angles of 50° and x° are on a straight line. Find x.
50 + x = 180 → x = 130°
Angles at a Point
All angles meeting at a single point add up to 360°.
Example: Three angles meet at a point: 90°, 120°, and y°. Find y.
90 + 120 + y = 360 → y = 150°
Vertically Opposite Angles
When two straight lines cross, the angles directly opposite each other are equal.
Angles in Parallel Lines
When a transversal cuts two parallel lines:
- Alternate angles (Z-angles) are equal. They are on opposite sides of the transversal between the parallel lines.
- Corresponding angles (F-angles) are equal. They are on the same side of the transversal, one between and one outside the parallel lines.
- Co-interior angles (C-angles or allied angles) add up to 180°. They are on the same side of the transversal between the parallel lines.
Interior Angles of a Polygon
Sum of interior angles = (n – 2) × 180°, where n is the number of sides.
Triangle (n=3): (3-2) × 180 = 180°
Quadrilateral (n=4): (4-2) × 180 = 360°
Pentagon (n=5): (5-2) × 180 = 540°
Hexagon (n=6): (6-2) × 180 = 720°
3. Properties of Triangles
Sum of Angles in a Triangle
The three angles in any triangle always add up to 180°.
Example: A triangle has angles of 47° and 68°. Find the third angle.
Third angle = 180 – 47 – 68 = 65°
Exterior Angle of a Triangle
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles (the two angles it is not next to).
Example: Two interior angles of a triangle are 40° and 75°. Find the exterior angle at the third vertex.
Exterior angle = 40 + 75 = 115°
Types of Triangles
By sides:
- Equilateral: all 3 sides equal, all angles = 60°
- Isosceles: 2 sides equal, 2 base angles equal
- Scalene: no sides equal, no angles equal
By angles:
- Acute triangle: all angles less than 90°
- Right-angled triangle: one angle exactly 90°
- Obtuse triangle: one angle greater than 90°
4. Pythagoras Theorem
Pythagoras theorem applies ONLY to right-angled triangles. It states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
Formula: a² + b² = c² (where c is the hypotenuse)
Finding the Hypotenuse
Example: A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
Finding a Shorter Side
Example: A ladder of length 13 m leans against a wall. Its foot is 5 m from the base of the wall. How high up the wall does it reach?
5² + h² = 13²
25 + h² = 169
h² = 144
h = 12 m
Pythagorean Triples (Common sets to know)
These are sets of three whole numbers that satisfy Pythagoras theorem. Knowing them saves time:
- 3, 4, 5 (3² + 4² = 5²)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
- Any multiple: 6, 8, 10 (doubles of 3,4,5) also works
5. Area and Perimeter of Plane Shapes
Rectangle
- Area = length × width
- Perimeter = 2(l + w)
Triangle
- Area = ½ × base × height
Example: A triangle has base 10 cm and perpendicular height 6 cm. Area = ½ × 10 × 6 = 30 cm²
Trapezium
- Area = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular height between them
Parallelogram
- Area = base × perpendicular height
Circle
- Area = πr² (use π = 22/7 or 3.142)
- Circumference = 2πr = πd
Example: Find the area of a circle with radius 7 cm. (Use π = 22/7)
Area = 22/7 × 7² = 22/7 × 49 = 154 cm²
Summary
- Angles on a straight line sum to 180°; angles at a point sum to 360°.
- Vertically opposite angles are equal.
- Alternate angles (parallel lines) are equal; co-interior angles sum to 180°.
- Sum of angles in any triangle = 180°.
- Exterior angle = sum of the two non-adjacent interior angles.
- Pythagoras: a² + b² = c² for right-angled triangles.
- Area of triangle = ½ × base × height.
- Area of circle = πr²; Circumference = 2πr.
Practice Questions
- Find the missing angle in a triangle with angles 58° and 74°.
- Two parallel lines are cut by a transversal. One alternate angle is 63°. Find the other alternate angle and also the co-interior angle on the same side.
- A right-angled triangle has a hypotenuse of 17 cm and one leg of 8 cm. Find the other leg.
- Find the area of a trapezium with parallel sides 9 cm and 13 cm and a height of 5 cm.
- A circle has diameter 14 cm. Find its circumference and area. (Use π = 22/7)
Answers: (1) 48° (2) Alternate = 63°; Co-interior = 117° (3) √(17²-8²) = √(289-64) = √225 = 15 cm (4) ½ × (9+13) × 5 = 55 cm² (5) Circumference = 44 cm; Area = 154 cm²
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