Edexcel IGCSE Maths 复习笔记 4.6.1 Circle Theorems - Angles at Centre & Circumference

What are circle theorems?

• You will have learned a lot of angle facts for your GCSE – angles in polygons, angles with parallel lines
• Circle Theorems deal with angle facts that occur with shapes and lines drawn within and connected to a circle

What do I need to know?

• Be familiar with the names of parts of a circle and lines within and outside of a circle:

• To solve hard problems you may need to use the angle facts you are already familiar with from triangles, polygons, and parallel lines
• You may also have to use circumference and area formulas, so ensure you’re familiar with: 

  C = πdand

  A = πr²

The first 3 circle theorems

1. The angle at the circumference in a semicircle is a right angle

• This theorem should be self-explanatory form its name/title
• The semicircle arises if you ignore the right-hand side of the diameter in the diagram above
• Look out for triangles hidden among other lines/shapes within the circle

2. Angles at the circumference subtended by the same arc are equal

• Subtended means the equal angles are created by drawing chords from the ends of the arc PQ
• Theses chords may or may not pass through the centre
• Both pairs of angles are equal

3. Angle subtended by an arc at the centre is twice the angle at the circumference

• Similar to above, the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the arc PQ
• This theorem can also happen when the ‘triangle parts’ overlap: