“Plane Geometry: Angles and Triangles”

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your WAEC exam.

Angle Basics:

Angle Type	Size
Acute	0° < θ < 90°
Right angle	θ = 90°
Obtuse	90° < θ < 180°
Straight	θ = 180°
Reflex	180° < θ < 360°
Complete	θ = 360°

Angle Relationships:

Relationship	Statement
Complementary	Two angles sum to 90°
Supplementary	Two angles sum to 180°
Vertically opposite	Vertically opposite angles are equal

Parallel Lines: When a transversal crosses parallel lines:

• Alternate angles are equal (Z-shape)
• Corresponding angles are equal (F-shape)
• Co-interior angles sum to 180° (C/U-shape)

⚡ WAEC Tip: For parallel lines, identify the angle relationship by the shape formed (Z, F, or C/U). Don’t just memorise — look at the diagram.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Triangle Angle Properties:

1. Sum of interior angles: $A + B + C = 180°$
2. Exterior angle: Equals the sum of the two opposite interior angles ($x = A + B$)
3. Angle bisector theorem: Bisector divides opposite side in ratio of adjacent sides

Triangle Types by Angles:

Type	Property	Example
Acute	All angles < 90°	50°, 60°, 70°
Right-angled	One angle = 90°	90°, 45°, 45°
Obtuse	One angle > 90°	100°, 40°, 40°

Triangle Types by Sides:

Type	Property
Equilateral	All sides equal, all angles 60°
Isosceles	Two sides equal, two angles equal (base angles)
Scalene	All sides different, all angles different

Isosceles Triangle:

• The angle opposite the equal sides are equal (base angles)
• Altitude from vertex bisects the base AND the angle
• Median from vertex also bisects the angle

Pythagoras Theorem: For a right-angled triangle with hypotenuse $c$: $$a^2 + b^2 = c^2$$

Pythagorean Triples (Memorise these):

• 3, 4, 5 (and multiples: 6,8,10; 9,12,15)
• 5, 12, 13 (and multiples: 10,24,26)
• 7, 24, 25
• 8, 15, 17
• 9, 40, 41

Worked Example: A ladder 13m long reaches a window 12m high. How far is the foot from the wall?

$c = 13$, $a = 12$, find $b$ $b^2 = c^2 - a^2 = 169 - 144 = 25$ $b = 5,\text{m}$

⚡ Common Mistake: Students apply Pythagoras to non-right-angled triangles. It ONLY works for right-angled triangles. Always check for the right angle first.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Congruence of Triangles: Two triangles are congruent (identical) if:

1. SSS: All three sides equal
2. SAS: Two sides and included angle equal
3. ASA: Two angles and included side equal
4. AAS: Two angles and a corresponding side equal
5. RHS: Right angle, hypotenuse, and one side equal (right-angled triangles)

Similar Triangles: Two triangles are similar if:

• All corresponding angles are equal
• All corresponding sides are in the same ratio
• Criteria: AA, SSS, SAS

Properties of Similar Triangles:

• Ratio of areas = (ratio of sides)²
• If scale factor is $k$, area scales by $k^2$

Worked Example: A pole 4m tall casts a shadow 6m long. At the same time, a tree casts a shadow 18m long. Find the tree height.

Since sun’s rays are parallel, triangles are similar. $\frac{\text{Pole}}{\text{Tree}} = \frac{\text{Pole shadow}}{\text{Tree shadow}}$ $\frac{4}{h} = \frac{6}{18}$ $h = \frac{4 \times 18}{6} = 12,\text{m}$

Area of a Triangle:

1. Using base and height: $A = \frac{1}{2} \times \text{base} \times \text{height}$

2. Heron’s formula (when all sides known): $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$

Example with Heron’s: Find area of triangle with sides 5cm, 6cm, 7cm.

$s = \frac{5+6+7}{2} = 9$ $A = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} ≈ 14.7,\text{cm}^2$

Concurrency in Triangles:

Line	Definition	Concurrency Point
Median	Joins vertex to midpoint of opposite side	Centroid
Altitude	Perpendicular from vertex to opposite side	Orthocentre
Angle bisector	Bisects an angle	Incentre
Perpendicular bisector	Bisects side at right angles	Circumcentre

The Centroid:

• Intersection of all three medians
• Divides each median in 2:1 ratio (vertex to centroid is 2/3 of median)

The Circumcentre:

• Centre of circle passing through all vertices (circumcircle)
• For obtuse triangle, lies outside the triangle
• For acute triangle, lies inside

The Incentre:

• Centre of circle inscribed within triangle (touches all sides)
• Equidistant from all sides

The Orthocentre:

• Intersection of all three altitudes
• For obtuse triangle, lies outside

Quadrilaterals:

Shape	Properties
Parallelogram	Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other
Rectangle	Parallelogram with all angles 90°; diagonals equal
Rhombus	Parallelogram with all sides equal; diagonals bisect at right angles
Square	Rectangle + Rhombus; all sides equal, all angles 90°
Trapezium	One pair of parallel sides
Kite	Two pairs of adjacent sides equal; one diagonal bisects the other at right angles

Circle Theorems:

1. Angle at centre is twice the angle at the circumference subtended by the same arc
2. Angles in the same segment are equal
3. Angle in a semicircle is a right angle
4. Opposite angles of a cyclic quadrilateral sum to 180°
5. Exterior angle equals interior opposite angle

⚡ WAEC Previous Year Pattern:

Year	Question	Concept
2023	Pythagoras theorem application	Right triangle
2022	Similar triangles	Scale factor
2021	Angle in semicircle	Circle theorem

Loci:

A locus is a set of points satisfying a condition.

Locus	Description
Points equidistant from A and B	Perpendicular bisector of AB
Points equidistant from lines L₁ and L₂	Angle bisector of L₁ and L₂
Points distance r from point P	Circle centred at P with radius r
Points distance d from line L	Two parallel lines distance d from L

Constructions (WAEC focus):

• Perpendicular bisector of a line segment
• Angle bisector
• Perpendicular from a point to a line
• Parallel line through a given point
• Triangle given SSS, SAS, ASA

⚡ Exam Strategy: Always draw a large, clear diagram. Label all known values. Write down the theorem you’re using (e.g., “angle in a semicircle = 90°”). Check that your answer is realistic for the geometry.

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