Ch 3 · 4.6 Angles · O Level Mathematics 4024
Topic 3 of 4 in Geometry — O Level Mathematics 4024 — 38 lessons.
Introduction
Roof trusses, tiled floors and road junctions all depend on angle facts. Exam questions ask not only for the angle but for the reason, using the correct geometrical words. This lesson lists the facts and the exact phrases to use. (Syllabus section 4.6.)
Key ideas — Basic angle facts
Base angles of an isosceles triangle are equal. Three-letter notation: angle ABC is the angle at B.
🧑🏫 Teacher's explanation
Think of angles as the amount of 'turning' you do at a corner. If you spin in a full circle and end up facing the same way, you have turned exactly 360^°. If you walk halfway around a circle on a flat floor, you have turned 180^°, which is what a straight line represents.
📖 Definitions
🔍 Worked example — Easy
In triangle ABC, angle A = 50^° and angle B = 70^°. Calculate angle C.
🔍 Worked example — Standard
Two parallel lines are intersected by a transversal. One of the co-interior angles is 115^°. Find the size of the other co-interior angle.
🔍 Worked example — Exam-style
A quadrilateral PQRS has angles P = 100^°, Q = 80^°, and R = 110^°. Calculate angle S.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Parallel lines
When a line crosses two parallel lines:
🧑🏫 Teacher's explanation
Imagine you are looking at a set of railway tracks stretching into the distance. The two rails are parallel lines. If a third rail crosses them at an angle, it creates several identical patterns at each intersection. Because the parallel lines never meet, the 'tilt' of the crossing line is exactly the same at both intersections.
📖 Definitions
🔍 Worked example — Easy
Two parallel lines are intersected by a transversal. If one of the corresponding angles is 75^°, find the value of its corresponding angle.
🔍 Worked example — Standard
In a diagram with two parallel lines, an angle of 110^° and an angle x are co-interior angles. Find x.
🔍 Worked example — Exam-style
Two parallel lines L_1 and L_2 are cut by a transversal. An angle ABC = 55^° is given. If angle BCD is an alternate angle to ABC, find angle BCD.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Polygons
For any polygon with n sides:
🧑🏫 Teacher's explanation
Imagine you are standing on one corner of a polygon and you walk along the perimeter. Every time you reach a corner, you turn by a certain amount to face the next side. By the time you return to your starting position, you have made one full rotation of 360^°. This total turn is the sum of all your exterior angles, regardless of how many sides the shape has.
📖 Definitions
🔍 Worked example — Easy
Calculate the sum of the interior angles of a hexagon.
🔍 Worked example — Standard
A regular octagon has interior angles of x. Calculate the value of x.
🔍 Worked example — Exam-style
An irregular pentagon has four interior angles measuring 100^°, 110^°, 120^°, and 90^°. Calculate the size of the fifth interior angle, y.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Calculate the size of each exterior angle. (b) Find the value of n. [3 marks]
Worked examples
Each exterior angle of a regular polygon is 24^°. n = 360 ÷ 24 = 15 sides; each interior angle is 156^°.
🔍 Worked example — Easy
A regular polygon has an exterior angle of 24^°. Calculate the number of sides and the size of each interior angle.
🔍 Worked example — Standard
A pentagon has interior angles of 100^°, 110^°, 95^°, 120^° and x. Calculate the value of x.
🔍 Worked example — Exam-style
Lines L_1 and L_2 are parallel. A transversal line intersects them. If one of the alternate angles is 65^°, find the size of the co-interior angle on the same side of the transversal.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 120^°. (2) 10 × 180^° = 1800^°. (3) exterior 18^°, so n = 20. (4) 70^°: base angles of an isosceles triangle are equal and the angles in a triangle add to 180^°.