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Plane Geometry and Circle Theorems

Mathematics · WAEC and JAMB · SS2 and SS3

Geometry questions are won by the reason, not only the number. WASSCE marking schemes award marks for naming the theorem used, so an answer with no reason given loses half the available credit even when the figure is correct.

What you need to know

  • Angles on a straight line add to 180 degrees, angles at a point add to 360 degrees, and vertically opposite angles are equal. These three settle most 'find the marked angle' questions.
  • When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180 degrees. Name which one you used.
  • The three angles of a triangle add to 180 degrees, and the exterior angle of a triangle equals the sum of the two interior opposite angles, which is far quicker than finding each angle separately.
  • In an isosceles triangle the base angles are equal, and the line from the apex to the midpoint of the base is perpendicular to the base. This fact converts many problems into right-angled triangles.
  • Pythagoras theorem applies only to right-angled triangles: the square on the hypotenuse equals the sum of the squares on the other two sides. The 3-4-5, 5-12-13 and 8-15-17 triples save calculation time.
  • The sum of the interior angles of a polygon with n sides is (n - 2) x 180 degrees, and the sum of the exterior angles of ANY convex polygon is always 360 degrees.
  • For a REGULAR polygon, each exterior angle is 360/n, and interior plus exterior angle always makes 180 degrees. Going through the exterior angle is the fast route to the number of sides.
  • The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining part of the circumference.
  • Angles in the same segment of a circle are equal, and the angle in a semicircle is a right angle.
  • Opposite angles of a cyclic quadrilateral add to 180 degrees, and the exterior angle of a cyclic quadrilateral equals the interior opposite angle.
  • A tangent is perpendicular to the radius at the point of contact, and the two tangents drawn from an external point are equal in length.
  • The angle between a tangent and a chord equals the angle in the alternate segment; this is the theorem candidates most often fail to recognise.
  • A line from the centre of a circle perpendicular to a chord bisects that chord, which creates a right-angled triangle linking the radius, half the chord and the distance from the centre.
  • Draw or redraw the figure large and mark every known angle on it as you find it. Working on a crowded printed diagram is where sign and label errors creep in.

Key terms

Transversal
A line that cuts across two or more other lines, creating corresponding, alternate and co-interior angle pairs.
Cyclic quadrilateral
A four-sided figure whose four vertices all lie on the circumference of the same circle.
Chord
A straight line joining any two points on the circumference of a circle.
Tangent
A straight line that touches a circle at exactly one point and is perpendicular to the radius at that point.
Alternate segment
The segment of a circle on the other side of a chord from the angle being considered.
Regular polygon
A polygon with all sides equal in length and all interior angles equal in size.
Exterior angle
The angle formed between one side of a polygon and the extension of the adjacent side.

Formulae

  • sum of interior angles of an n-sided polygon = (n - 2) x 180 degrees
  • sum of exterior angles of any convex polygon = 360 degrees
  • each exterior angle of a regular n-gon = 360/n
  • each interior angle of a regular n-gon = 180 - 360/n
  • Pythagoras: hypotenuse^2 = a^2 + b^2
  • angle at centre = 2 x angle at circumference on the same arc
  • opposite angles of a cyclic quadrilateral sum to 180 degrees
  • tangent length from external point: t = sqrt(d^2 - r^2), where d is the distance from the centre

Worked examples

Each interior angle of a regular polygon is 150 degrees. Find the number of sides and the sum of its interior angles.

  1. Interior and exterior angles are supplementary, so each exterior angle = 180 - 150 = 30 degrees.
  2. The exterior angles of any polygon add to 360 degrees, so the number of sides n = 360/30 = 12.
  3. The sum of the interior angles is (n - 2) x 180 = (12 - 2) x 180.
  4. That gives 10 x 180 = 1800 degrees.
  5. Check: 1800 divided by 12 sides = 150 degrees per angle, as stated.

PQRS is a cyclic quadrilateral in which angle P = (3x + 10) degrees and angle R = (2x + 20) degrees. Find x and hence the size of each of the two angles.

  1. Angles P and R are opposite angles of a cyclic quadrilateral, so they add to 180 degrees.
  2. Form the equation: (3x + 10) + (2x + 20) = 180.
  3. Collect terms: 5x + 30 = 180.
  4. Subtract 30: 5x = 150, so x = 30.
  5. Angle P = 3(30) + 10 = 100 degrees.
  6. Angle R = 2(30) + 20 = 80 degrees.
  7. Check: 100 + 80 = 180 degrees, as required.

A tangent TA is drawn from an external point T to a circle of centre O and radius 5 cm. If OT = 13 cm, find the length of the tangent TA.

  1. The tangent is perpendicular to the radius at the point of contact, so angle OAT = 90 degrees.
  2. Triangle OAT is therefore right-angled with hypotenuse OT = 13 cm and one side OA = 5 cm.
  3. By Pythagoras: TA^2 = OT^2 - OA^2 = 13^2 - 5^2.
  4. That is 169 - 25 = 144.
  5. So TA = sqrt(144) = 12 cm.
  6. Note this is the 5-12-13 triple, a useful check.

The mistake to avoid

Candidates state the right angle but give no reason, and the marking scheme withholds the reasoning mark. The other frequent error is assuming a diagram is to scale and measuring an angle with a protractor instead of calculating it; printed WASSCE figures are deliberately not drawn to scale.

In the exam

Write the reason beside every step in short form, such as 'angles in the same segment' or 'tangent perpendicular to radius', because each named theorem earns its own mark. Mark all equal sides and equal angles on your own copy of the figure. If a question says 'prove', finish with a clear concluding line stating what has been shown.