Recognise the angles formed inside a circle

Key Rules:

• 1. The angle at the centre is double the angle at the circumference standing on the same arc, so an angle in a half circle, standing on a diameter, is exactly 90 degrees.
• 2. Angles at the circumference standing on the same arc are equal, and opposite angles of a four-sided shape with all four corners on the circle add to 180 degrees.
• 3. The angle between a tangent and a chord equals the angle at the circumference in the alternate segment, the part of the circle on the other side of that chord.

Remember:

• •Find the angle, then name the rule that gave it. In a Venn diagram, fill the overlap first.

Don’t Forget...

• •Halving the marked angle at the centre when it does not stand on the same arc as the angle you want. The doubling rule pairs a centre angle and a circumference angle standing on the same arc; an angle standing on the other arc pairs with the reflex angle at the centre instead.
• •Using the 180 degree rule for opposite angles in a four-sided shape that has one corner at the centre. The rule needs all four corners on the circumference; a corner at the centre is not on the circle, and that shape is handled by the doubling rule instead.
• •Writing 18 and 14 straight into the French-only and Spanish-only regions of a Venn diagram. Each circle's total already includes the overlap, so the French-only region holds 18 minus the overlap; filling the overlap first keeps every value counted exactly once.

Worked Example:

Points A, B, C and D lie on a circle with centre O, in that order around the circle. The angle AOC at the centre, on the side nearest D, is 116 degrees. Work out angle ABC and angle ADC, giving a reason for each.

Angle ABC at the circumference stands on the same arc AC as the marked angle at the centre, so it is half of 116, which is 58 degrees, because the angle at the centre is double the angle at the circumference.

ABCD has all four corners on the circle, so its opposite angles add to 180 degrees.

Angle ADC = 180 - 58 = 122 degrees.

Check: the reflex angle AOC is 360 - 116 = 244 degrees, and half of 244 is 122, which confirms angle ADC.

Answer: Angle ABC = 58 degrees and angle ADC = 122 degrees

Why It Works:

Every radius of a circle is the same length, so joining the centre to points on the circumference creates isosceles triangles with equal base angles. Splitting the angle at the centre into two such triangles and adding their base angles proves the doubling rule, and the half circle, same arc and cyclic four-sided-shape rules all follow from it, so one chain of reasoning underwrites the whole set.

Another Example:

In a class of 30 students, 18 study French, 14 study Spanish and 5 study neither language. Use a Venn diagram to find how many students study both.

The two circles together must hold 30 - 5 = 25 students.

Adding the circle totals counts the overlap twice: 18 + 14 = 32.

The double-counted amount is the overlap: 32 - 25 = 7, so 7 students sit in both circles.

Fill the regions to check: French only is 18 - 7 = 11, Spanish only is 14 - 7 = 7, and 11 + 7 + 7 + 5 = 30.

Answer: 7 students study both languages