This note covers the angle facts for lines, triangles, quadrilaterals and parallel lines, the angles of polygons, and the basic properties of circles that do not need the full circle theorems. These facts are the starting point for almost every geometry problem, so you need to recall them quickly and give a reason for each step.
Key ideas
Basic angle facts.
- Angles on a straight line add up to \(180^\circ\).
- Angles around a point add up to \(360^\circ\).
- Vertically opposite angles (formed where two straight lines cross) are equal.
- The angles in a triangle add up to \(180^\circ\). In an isosceles triangle, the two base angles are equal. In an equilateral triangle, all angles are \(60^\circ\).
- The angles in any quadrilateral add up to \(360^\circ\).
- An exterior angle of a triangle equals the sum of the two interior opposite angles.
Parallel lines. When a straight line (a transversal) crosses two parallel lines:
- Corresponding angles are equal (they sit in matching positions, forming an F shape).
- Alternate angles are equal (they lie on opposite sides of the transversal, forming a Z shape).
- Co-interior angles add up to \(180^\circ\) (they lie between the parallel lines on the same side, forming a C shape).
Polygons. A polygon is a closed shape with straight sides. A regular polygon has equal sides and equal angles. For a polygon with \(n\) sides:
- The sum of the interior angles is \((n-2) \times 180^\circ\).
- The sum of the exterior angles is always \(360^\circ\).
- Each interior angle and its exterior angle add up to \(180^\circ\).
- For a regular polygon, each exterior angle is \(\dfrac{360^\circ}{n}\) and each interior angle is \(180^\circ - \dfrac{360^\circ}{n}\).
The sum formula comes from splitting the polygon into \(n-2\) triangles by drawing diagonals from one corner.
Types of triangle and quadrilateral. You should know the names and properties of the isosceles, equilateral and right-angled triangles, and the square, rectangle, parallelogram, rhombus, kite and trapezium. Parallelograms have opposite angles equal, and adjacent angles that add up to \(180^\circ\). A kite has one pair of equal opposite angles and its diagonals cross at right angles.
Circle vocabulary. The centre, radius, diameter (twice the radius), circumference, chord (a straight line joining two points on the circle), arc, sector, segment and tangent (a line touching the circle at exactly one point).
Basic circle properties.
- A tangent is perpendicular to the radius at the point of contact.
- Two tangents drawn from the same outside point are equal in length.
- A line from the centre perpendicular to a chord bisects the chord (cuts it in half). This also means that the line from the centre to the midpoint of a chord is perpendicular to the chord.
- A triangle formed by two radii and a chord is isosceles.
The longer circle theorems (angle at the centre, angles in the same segment, cyclic quadrilaterals and the alternate segment theorem) are covered separately at Higher tier.
Giving reasons. In angle questions, state the fact you are using in words, for example "angles in a triangle add up to \(180^\circ\)" or "alternate angles are equal". Say which fact each line of working relies on.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Angles on a line | Sum to \(180^\circ\). |
| Angles at a point | Sum to \(360^\circ\). |
| Interior angle sum | \((n-2) \times 180^\circ\) for an \(n\)-sided polygon. |
| Exterior angle sum | \(360^\circ\) for any polygon. |
| Regular polygon exterior angle | \(\dfrac{360^\circ}{n}\). |
| Alternate / corresponding angles | Equal when the lines are parallel. |
| Co-interior angles | Add up to \(180^\circ\) when the lines are parallel. |
| Tangent and radius | Meet at \(90^\circ\). |
| Chord bisector | Line from the centre perpendicular to a chord halves it. |
Worked examples
Example 1 Each interior angle of a regular polygon is \(156^\circ\). (a) Find the number of sides. (b) Find the sum of the interior angles.
(a) The exterior angle is \(180^\circ - 156^\circ = 24^\circ\).
The exterior angles sum to \(360^\circ\), so \(n = \dfrac{360}{24} = 15\).
(b) Sum of interior angles \(= (15 - 2) \times 180^\circ = 13 \times 180^\circ = 2340^\circ\).
Check: \(15 \times 156^\circ = 2340^\circ\). Correct.
Answer: (a) 15 sides (b) \(2340^\circ\)
Example 2 \(ABCD\) is a trapezium with \(AB\) parallel to \(DC\). Angle \(DAB = 78^\circ\), angle \(ABC = (2x + 14)^\circ\) and angle \(BCD = (3x - 4)^\circ\). Find \(x\), and all four angles.
Because \(AB \parallel DC\), the angles at \(B\) and \(C\) lie between the parallel lines on the same side of \(BC\). They are co-interior, so they add up to \(180^\circ\).
\((2x + 14) + (3x - 4) = 180\), so \(5x + 10 = 180\), which gives \(5x = 170\) and \(x = 34\).
Angle \(ABC = 2(34) + 14 = 82^\circ\). Angle \(BCD = 3(34) - 4 = 98^\circ\).
Angles at \(A\) and \(D\) are also co-interior (between \(AB\) and \(DC\), on the same side of \(AD\)): angle \(ADC = 180^\circ - 78^\circ = 102^\circ\).
Check with the quadrilateral sum: \(78 + 82 + 98 + 102 = 360\). Correct.
Answer: \(x = 34\); angles \(78^\circ, 82^\circ, 98^\circ, 102^\circ\)
Example 3 A circle has centre \(O\) and radius 10 cm. A chord \(AB\) has length 16 cm. Find the perpendicular distance from \(O\) to the chord. Then find the length of the tangent from a point \(P\) that is 26 cm from \(O\).
The perpendicular from \(O\) to \(AB\) bisects the chord. Let \(M\) be the midpoint, so \(AM = 8\) cm.
Triangle \(OMA\) is right-angled at \(M\) with hypotenuse \(OA = 10\) (a radius).
\(OM^2 = 10^2 - 8^2 = 100 - 64 = 36\), so \(OM = 6\) cm.
For the tangent, let \(T\) be the point of contact. The radius \(OT\) is perpendicular to the tangent \(PT\), so triangle \(OTP\) is right-angled at \(T\) with hypotenuse \(OP = 26\).
\(PT^2 = 26^2 - 10^2 = 676 - 100 = 576\), so \(PT = 24\) cm.
Answer: distance 6 cm; tangent length 24 cm
Common errors
- Common error: confusing alternate and co-interior angles. Alternate angles are equal, while co-interior angles add up to \(180^\circ\).
- Common error: using \(n \times 180^\circ\) instead of \((n-2) \times 180^\circ\) for the interior angle sum.
- Common error: using the interior angle in \(\dfrac{360}{\text{angle}}\) to find the number of sides. It must be the exterior angle.
- Common error: stating an angle without a reason when the question says "give reasons".
- Common error: forgetting that the tangent meets the radius at \(90^\circ\), so a right-angled triangle is missed.
Exam tip: Mark every angle you find on the diagram as you go, and write the reason beside it. A later step often needs an angle that you found earlier.
Check yourself
- Find the sum of the interior angles of a decagon (10 sides), and the size of each interior angle if it is regular.
Answer
Sum \(= (10-2) \times 180^\circ = 1440^\circ\). Each angle of a regular decagon \(= 1440^\circ \div 10 = 144^\circ\). Check: the exterior angle is \(36^\circ\) and \(180 - 36 = 144\). - Triangle \(PQR\) has \(PQ = PR\) and angle \(QPR = 40^\circ\). Find angle \(PQR\).
Answer
The base angles are equal. Each is \((180^\circ - 40^\circ) \div 2 = 70^\circ\). - A tangent \(PT\) touches a circle with centre \(O\) at \(T\). Angle \(TPO = 25^\circ\). Find angle \(TOP\).
Answer
Angle \(OTP = 90^\circ\) (tangent and radius). Then angle \(TOP = 180^\circ - 90^\circ - 25^\circ = 65^\circ\). - Two parallel lines are cut by a transversal. One angle is \(112^\circ\). Find the angle that is co-interior to it, and the angle that is alternate to it.
Answer
Co-interior: \(180^\circ - 112^\circ = 68^\circ\). Alternate angle: it is equal to \(112^\circ\).