angles in circle

Formulas for Working with Angles in Circles
(Intercepted arcs are arcs “cut off” or “lying between” the sides of the specified angles.)

There are basically five circle formulas that
you need to remember:               
 
1.  Central Angle:  
A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.
 

Central Angle = Intercepted Arc

<AOB is a central angle.
 Its intercepted arc is the minor arc from A to B.  m<AOB = 80º
 

2.  Inscribed Angle:
An inscribed angle is an angle with its vertex “on” the circle, formed by two intersecting chords.

Inscribed Angle = Intercepted Arc

<ABC is an inscribed angle.
 Its intercepted arc is the minor arc from A to C.
m<ABC = 50º

          
3.  Tangent Chord Angle:
An angle formed by an intersecting tangent and chord has its vertex “on” the circle.

Tangent Chord Angle =
 Intercepted Arc

<ABC is an angle formed by a tangent and chord.
Its intercepted arc is the minor arc from A to B.
m<ABC = 60º
 

4.  Angle Formed Inside of a Circle by Two Intersecting Chords:
When two chords intersect “inside” a circle, four angles are formed.  At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture.  Remember:  vertical angles are equal.

Angle Formed Inside by Two Chords =
 
Sum of Intercepted Arcs 
Once you have found ONE of these angles, you automatically know the sizes of the other three by using your knowledge of vertical angles (being equal) and adjacent angles forming a straight line (adding to 180).

 


<BED is formed by two intersecting chords.
 Its intercepted arcs are .
  [Note:  the intercepted arcs belong to the set of vertical angles.]
 
also, m<CEA = 120º (vetical angle)
m<BEC and m<DEA = 60º by straight line.
5.  Angle Formed Outside of a Circle by the Intersection of:
“Two Tangents” or “Two Secants” or “a Tangent and a Secant”.

The formulas for all THREE of these situations are the same:
Angle Formed Outside = Difference of Intercepted Arcs 
(When subtracting, start with the larger arc.)
Two Tangents:
<ABC is formed by two tangents intersecting outside of circle O. 
The intercepted arcs are minor arc AC and major arc AC.  These two arcs together comprise the entire circle.
Special situation for this set up:  It can be proven that <ABC and central <AOC are supplementary.  Thus the angle formed by the two tangents and its first intercepted arc also add to 180º.
Two Secants:
<ACE is formed by two secants intersecting outside of circle O. 
The intercepted arcs are minor arcs BD and AE. 
a Tangent and a Secant:
<ABD is formed by a tangent and a secant intersecting outside of circle O. 
The intercepted arcs are minor arcs AC and AD. 

Circle Theorems

Circles

A circle is a set of points which are all a certain distance from a fixed point known as the centre.
A line joining the centre of a circle to any of the points on the circle is known as a radius.

The circumference of a circle is the length of the circle. The circumference of a circle = 2 × π × the radius.

a sector, an arc and chord

The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.

Theorems

Angles Subtended on the Same Arc

Angles subtended on the same arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angle in a Semi-Circle

angle in a semi-circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.

This proof is higher tier Proof

We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

Divide the triangle in two

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

Two isosceles triangles

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.

Tangents

A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).

A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.

angle with a tangent

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Tangents from an external point are equal in length

Angle at the Centre

Angle at the centre

The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

This proof is higher tier Proof

You might have to be able to prove this fact:

proof diagram 1

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b

proof diagram 2
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a
Similarly, ∠BOX = 180 – 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX
= 360 – (180 – 2a) – (180 – 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB

This section is higher tier Alternate Segment Theorem

Alternate segment theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.

Proof

You may have to be able to prove the alternate segment theorem:

proof of alternate segment theorem

We use facts about related angles:

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.

This proof is higher tier Area of Sector and Arc Length

A sector

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360

 by,

Wong Swan Swan

 

 

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