Seven facts unlock almost every angle question: angles at a point, on a straight line, vertically opposite, in a triangle, in a quadrilateral, and the three parallel-line rules. The marks come from naming the fact you used.
Angles at a point add to \(360^\circ\); on a straight line to \(180^\circ\); vertically opposite angles are equal; a triangle's angles add to \(180^\circ\) and a quadrilateral's to \(360^\circ\). With parallel lines, corresponding and alternate angles are equal and co-interior angles add to \(180^\circ\).
Fig. 1: a transversal crosses two parallel lines at 65° (drawn to scale). The 65° at Q is corresponding to the 65° above P, alternate to the 65° below-left of P, and co-interior with the 115° at P (65 + 115 = 180)
Section 01
Points, lines and vertically opposite.
In the angle facts tab, drag the lines and watch the angle readouts. Check that the angles on a straight line always total 180°, the angles round a point total 360°, and the opposite angles where two lines cross stay equal.
Angles at a point add up to \(360^\circ\).
Angles on a straight line (at a point on the line) add up to \(180^\circ\).
Vertically opposite angles are equal: where two straight lines cross, the angles facing each other match.
Three-letter notation: angle ABC is the angle AT B, between BA and BC. The middle letter is the vertex.
Worked example On a straight line
Three angles on a straight line are \(3x\), \(2x\) and \(40^\circ\). Find \(x\).
Angles on a straight line add to \(180^\circ\): \(3x + 2x + 40 = 180\).
\(5x = 140\).
\(x = 28\).
Answer: \(x = 28\) (the angles are \(84^\circ\), \(56^\circ\), \(40^\circ\))
Worked example At a point
Four angles meet at a point: \(120^\circ\), \(90^\circ\), \(x\) and \(2x\). Find \(x\).
Angles at a point add to \(360^\circ\): \(120 + 90 + x + 2x = 360\).
\(3x = 150\).
\(x = 50\).
Answer: \(x = 50^\circ\)
Examiner trap
Write the reason in words, using the proper term: 'angles on a straight line add to 180°'. A correct number with no reason, or with 'because it is', often loses the reason mark.
Section 02
Parallel lines.
In the parallel lines tab, drag the transversal to change its angle. Notice which angles stay equal (corresponding, the F shape; alternate, the Z shape) and which pairs always add to 180° (co-interior, the C shape). Compare with Fig. 1.
Corresponding angles are equal: same position at each crossing (an F shape).
Alternate angles are equal: between the parallel lines, on opposite sides of the transversal (a Z shape).
Co-interior angles add up to \(180^\circ\): between the parallel lines, on the same side (a C or U shape).
In exams, use the proper names. 'F angles' or 'Z angles' are not accepted as reasons.
Worked example Co-interior angles
Two parallel lines are crossed by a transversal. Two co-interior angles are \(2x + 15^\circ\) and \(3x + 15^\circ\). Find \(x\) and both angles.
\(5x + 30 = 180\), so \(5x = 150\) and \(x = 30\).
The angles are \(2(30) + 15 = 75^\circ\) and \(3(30) + 15 = 105^\circ\). Check: \(75 + 105 = 180\).
Answer: \(x = 30\); angles \(75^\circ\) and \(105^\circ\)
Your turn
In Fig. 1, explain why the angle below-left of P is \(65^\circ\), using a parallel-line reason.
It is alternate to the \(65^\circ\) angle at Q (between the parallel lines, on opposite sides of the transversal), and alternate angles are equal.
Examiner trap
Co-interior angles are NOT equal: they add to \(180^\circ\). Setting \(2x + 15 = 3x + 15\) gives \(x = 0\), a sure sign the wrong rule was used.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
What do the angles on a straight line add up to?
Quick check+10 XP
Two straight lines cross. One angle is \(70^\circ\). What is the vertically opposite angle?
Quick check+10 XP
Alternate angles between parallel lines are
Quick check+10 XP
What do the angles of a quadrilateral add up to?
Section 03
Triangles and quadrilaterals.
In the triangles and quadrilaterals tab, drag the vertices to any position. The angle total of the triangle never changes from 180°, and of the quadrilateral from 360°, because a quadrilateral splits into two triangles.
Angle sum of a triangle \(= 180^\circ\). Angle sum of a quadrilateral \(= 360^\circ\).
Isosceles triangle: the two base angles (opposite the equal sides) are equal. Equilateral: every angle is \(60^\circ\).
Exterior angle of a triangle = the sum of the two interior angles opposite it (it makes \(180^\circ\) with the third angle, and so do those two together).
Worked example A quadrilateral
The angles of a quadrilateral are \(x\), \(2x\), \(3x\) and \(90^\circ\). Find the largest angle.
Angle sum of a quadrilateral is \(360^\circ\): \(x + 2x + 3x + 90 = 360\).
\(6x = 270\), so \(x = 45\).
Largest angle \(= 3x = 135^\circ\).
Answer: \(135^\circ\)
Your turn
Triangle ABC is isosceles with AB = AC and angle BAC = \(36^\circ\). Find angle ABC.
Base angles are equal: \((180^\circ - 36^\circ) \div 2 = 72^\circ\).
Examiner trap
In an isosceles triangle, the equal angles are the ones OPPOSITE the equal sides, not the angle between them. Check which sides are marked equal before you halve anything.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
Three angles on a straight line are \(2x\), \(3x\) and \(x + 30^\circ\). Find \(x\).
Exam style+20 XP
A transversal crosses two parallel lines. Two co-interior angles are \(3x + 10^\circ\) and \(2x + 20^\circ\). Find \(x\).
Exam style+20 XP
Triangle ABC has AB = AC and angle ABC = \(68^\circ\). Find angle BAC.
Exam style+20 XP
Two interior angles of a triangle are \(47^\circ\) and \(61^\circ\). Find the exterior angle at the third vertex.
Skill unlocked
Angles, with reasons.
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