Mathematics · Lesson 14

Plane geometry: angles, polygons, triangles, congruence & similarity

JAMBWAECNECO

What the syllabus asks

JAMB: plane geometry: angles and lengths, triangles and polygons, congruence and similarity

WAEC: angles at a point and on a line, angles formed by parallel lines and a transversal, angle properties of triangles and polygons (interior and exterior angles), properties of quadrilaterals, congruent triangles, similar triangles, mid-point and intercept theorems, deductive proofs

NECO: angles at a point and on a line, angles formed by parallel lines and a transversal, angle properties of triangles and polygons (interior and exterior angles), properties of quadrilaterals, congruent triangles, similar triangles, mid-point and intercept theorems, deductive proofs

What you will learn

Name and calculate angles at a point, on a line and vertically opposite.
Use corresponding, alternate and co-interior angles with parallel lines.
Apply the angle sum and exterior angle properties of triangles.
State and use the properties of special triangles and quadrilaterals.
Calculate interior and exterior angles of polygons, including regular polygons.
Prove triangles congruent using SSS, SAS, ASA/AAS and RHS.
Use similar triangles to find lengths and areas, including the intercept and mid-point theorems.

Key points

Complementary angles add to 90°, supplementary to 180°.
Straight line 180°, point 360°, vertically opposite angles equal.
Parallel lines: corresponding (F) and alternate (Z) equal; co-interior (C) add to 180°.
Triangle: angles add to 180°; exterior angle = sum of interior opposite angles.
Isosceles: angles opposite equal sides are equal; equilateral: all 60°.
Quadrilateral angles add to 360°; know parallelogram, rhombus, rectangle, square, kite properties.
Polygon interior sum (n − 2)180°; exterior sum 360°; regular: exterior = 360°/n.
Congruence: SSS, SAS (included angle), ASA/AAS, RHS; not AAA or SSA.
Similar triangles: equal angles, proportional sides; compare with the WHOLE side.
Area ratio = (length ratio)²; intercept theorem AD/DB = AE/EC; mid-point line = half the third side.

Start here: Angles and their names

Geometry is the study of shape and space. It is one of the oldest branches of mathematics: builders of the walls of ancient Kano and the farmers who marked out plots along the Niger needed right angles and straight lines long before textbooks existed.

An angle is the amount of turning between two lines that meet at a point (the vertex). We measure it in degrees (°); a full turn is 360°.

Name	Size	Example
Acute	less than 90°	35°
Right	exactly 90°	the corner of an exercise book
Obtuse	between 90° and 180°	120°
Straight	exactly 180°	a straight line
Reflex	between 180° and 360°	250°

Two angles are complementary if they add up to 90°, and supplementary if they add up to 180°. A trick to remember: C comes before S in the alphabet, and 90 comes before 180.

Worked example 14.1

Find (a) the complement of 37°, (b) the supplement of 37°, (c) the angle whose supplement is four times its complement.

(a) 90 − 37 = 53°. (b) 180 − 37 = 143°.
(c) Let the angle be x. Supplement = 180 − x; complement = 90 − x.
180 − x = 4(90 − x) → 180 − x = 360 − 4x → 3x = 180 → x = 60°.
Check: supplement 120°, complement 30°, and 120 = 4 × 30 ✓.

53°, 143°, 60°

⚠️ Common mistake: mixing up complementary (90°) and supplementary (180°).

Try it yourself. Two complementary angles are in the ratio 2 : 3. Find them.

Show solution

90 ÷ 5 = 18; angles 36° and 54°.

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Practice questions

1. Angles at a point are 2x, 3x, 4x and 90°. Find x.

Show worked answer

9x = 270 → x = 30°.

2. In ΔABC, ∠A = 52° and ∠B = 71°. Find the exterior angle at C.

Show worked answer

52 + 71 = 123°.

3. Each interior angle of a regular polygon is 162°. Find n.

Show worked answer

Exterior 18°; n = 20.

4. Find the sum of the interior angles of a heptagon.

Show worked answer

5 × 180 = 900°.

5. In isosceles ΔXYZ, XY = XZ and ∠Y = 52°. Find ∠X.

Show worked answer

180 − 104 = 76°.

Key words

Acute angle: An angle between 0° and 90°.
Obtuse angle: An angle between 90° and 180°.
Reflex angle: An angle between 180° and 360°.
Complementary angles: Two angles that add up to 90°.
Supplementary angles: Two angles that add up to 180°.
Transversal: A line that cuts across two or more lines.
Corresponding angles: Angles in matching positions at the two intersections (an F shape).
Alternate angles: Angles on opposite sides of the transversal, between the parallels (a Z shape).
Co-interior angles: Angles on the same side of the transversal between the parallels (a C/U shape); they add to 180°.
Polygon: A closed plane figure with straight sides.
Regular polygon: A polygon with all sides and all angles equal.
Congruent: Identical in shape and size.

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