Step by step
The question
Question: find the angle x at the center O.
PA and PB are tangents from P.
The angle at P is 40°.
1 Use the tangent rule
PA and PB touch the circle at A and at B.
Both are tangents from the point P.
Join the center O to A and to B.
OA and OB are both radii, so they are equal.
Each radius meets its tangent at a right angle.
2 Use the angle fact
O, A, P and B make a shape with 4 sides.
The dashed line cuts it into 2 triangles: 2 times 180° is 360°.
So the four angles add up to 360: 90 + 90 + 40 + x = 360.
Careful!
Do not forget the two right angles at A and B.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
90 + 90 + 40 = 220.
Now we have 220 + x = 360.
4 Remove the 220
We want x by itself.
The 220 is in the way.
To remove 220, we take away 220.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
220 and − 220 make zero.
Cross them out.
360 − 220 = 140.
So x = 140!
5 Check
Let's check: put x = 140 back and add the four angles.
They add up to 360°.
So x = 140 is right!
Let's look back.
First: use the tangent rule.
Then: use the angle fact.
Then: group like terms.
Then: remove the 220.
Last: check.