Circles: Angle Properties — Part 1: Angle at the centre, same segment and semicircle: visual lesson, ICSE Class 10 Mathematics

Lesson

Circles: angles, Part 1

An arc of a circle looks twice as wide from the centre as it does from the rim. In this part we prove that, and use it together with two companion facts to find angles.

In this lesson you will

State and prove that the angle at the centre is double the angle at the circumference
Use equal chords and equal arcs with the angles they subtend at the centre
Use angles in the same segment to transfer an angle across a circle figure
Prove and use the fact that the angle in a semicircle is a right angle

The arc $AB$ looks twice as wide from the centre ($100^\circ$) as from the rim ($50^\circ$).

Concept

What you will be able to do

State and prove the centre and circumference theorem
Use equal chords and equal arcs
Use angles in the same segment
Prove and use the angle in a semicircle

Five properties; part 1 covers the first four.
Result	In words	Where
Angle at the centre	double the angle at any point of the remaining circumference	proved and used in part 1
Equal chords and arcs	equal angles at the centre	used in part 1
Angles in the same segment	equal to each other	used in part 1
Angle in a semicircle	a right angle	proved and used in part 1
Cyclic quadrilateral	opposite angles supplementary; exterior angle equals interior opposite	part 2

Concept

Words about a circle

A chord joins two points of the circle; a chord through the centre is a diameter. A chord cuts the circle into a minor arc and a major arc. An arc subtends an angle at a point when its ends are joined to that point.

the two parts cut off by a chord are the segments
a point on the major arc and the centre look at the minor arc

Diameter $AB$, chord $PQ$ and radius $OR$. The chord $PQ$ cuts the circle into a minor arc and a major arc.

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