Angles and Circle Theorems

The rules that angles obey

Angles are not guessed, they are worked out. A handful of rules, applied one at a time with a reason for each, will unlock almost any angle in a diagram. This module builds up in two parts: - The basic rules: angles on a straight line add to a half turn, angles at a point add to a full turn, and the angles in a triangle add to a half turn. - The circle theorems: a special set of rules for angles drawn inside and around a circle. Pythagoras and trigonometry, the other side of geometry, are handled in a separate module. Here the focus is angles and the circle theorems, and above all giving a reason at every step.

Words for angles and circles

Five terms you need before you use them. Learn what each one means.

Match each circle theorem to what it states

Angle at the centre against angle at the circumference

One of the most useful circle theorems links two angles standing on the same arc. Keeping them straight is the key.

Which theorem applies?

A triangle is drawn inside a circle so that one of its sides passes through the centre as a diameter. What is the angle opposite that diameter?

• A right angle, because the angle in a semicircle is ninety degrees
• An angle of sixty degrees
• It cannot be worked out from this information
• The same size as the angle at the centre

Tap the two true facts about a tangent

Tap the TWO statements that are true of a tangent to a circle.

• It touches the circle at exactly one point
• It meets the radius at that point at a right angle
• It always passes straight through the centre
• It joins two separate points on the circumference

Working an angle problem the safe way

Angle questions reward a calm, orderly method far more than a lucky guess. The same routine works every time. First mark on every angle you are told. Then name the rule that connects what you know to what you want: angles on a line add to a half turn, angles at a point add to a full turn, the angles in a triangle add to a half turn, or one of the circle theorems. Then work one step at a time, and beside each step write the reason. In the exam the marks are split between the correct number and the reason given for it, so an answer with no reasons throws half the marks away. Slow and reasoned beats fast and bare.

Pick the true angle facts

Select the TWO statements that are true.

• The angles inside a triangle add up to a half turn, 180 degrees
• The angles around a single point add up to 360 degrees
• The angle in a semicircle is always sixty degrees
• Opposite angles of a cyclic quadrilateral are always equal

Order how to solve an angle problem

Put the steps of a reliable method for an angle problem in order.

• Mark on every angle you are given
• Identify which angle rule or theorem applies
• Work out the unknown angle step by step
• Write the reason beside each step
• State the final angle clearly with its units

Reading a circle, one step at a time

Here is how to read a circle problem calmly, one rule at a time.\n\nWhat is given: a point sits at the middle of a circle, and two lines run from it out to the edge, making an angle of eighty degrees between them at the middle. The first rule: the angle made out at the edge, standing on the same arc, is half the angle made at the middle. Half of eighty is forty, so the angle at the edge is forty degrees. A second point: another angle drawn at the edge, standing on that very same arc, must match it, so it is forty degrees as well. The habit that scores: at every step, name the rule you used. The marks come not only from the number but from the reason written beside it. Read the circle rule by rule, and even a crowded diagram comes apart neatly.

Complete the circle facts

A straight line that touches a circle at exactly one point is a _____. A four-sided shape with all corners on the circle is a cyclic _____. The angle drawn in a _____ is always a right angle. The angle at the centre is _____ the angle at the circumference on the same arc.

Build an angle reason

Choose the words that complete this statement about circle angles.

Work out the angle at the circumference

The reflex angle at the centre of a circle is 260 degrees. Angles at a point add up to 360 degrees, so first take the reflex angle from 360 to find the ordinary angle at the centre. The angle at the circumference on the same arc is half the angle at the centre. What is the angle at the circumference, in degrees?

Pick the right theorem

Read each case and choose the theorem that settles it, then think about why.

• A triangle drawn inside a circle has one side that is a diameter. What can you say about the angle opposite that side?
• A four-sided shape is drawn with all four corners on a circle. What is true of a pair of opposite angles?
• A tangent touches a circle, and a radius is drawn to the point where it touches. What is the angle between the tangent and the radius?

Explain the main circle theorems

A student in the year below cannot remember the circle theorems. Explain them clearly using what this module has covered.

• State the angle in a semicircle theorem
• State the rule linking the angle at the centre and the angle at the circumference
• State what is true of angles in the same segment
• State the cyclic quadrilateral rule
• Finish by explaining why you must give a reason at every step