Step by step
The question
Question: find x.
PT is a tangent to the circle with center O.
The angle at O is x.
The angle at P is 35°.
1 Use the tangent rule
PT touches the circle at T, and nowhere else.
It is called a tangent.
OT goes from the center to the touch point: a radius.
The radius meets the tangent at a right angle: 90°.
This is always true for a tangent.
2 Use the angle fact
O, T and P make a triangle.
Its three angles add up to 180°.
So x + 35 + 90 = 180.
Careful!
The right angle at T is one of the three angles.
It goes in the sum too.
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.
35 + 90 = 125.
Now we have x + 125 = 180.
4 Remove the + 125
We want x by itself.
The + 125 is in the way.
To remove + 125, we take away 125.
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.
+ 125 and − 125 make zero.
Cross them out.
180 − 125 = 55.
So x = 55!
5 Check
Let's check: put x = 55 back and add the three angles.
They add up to 180°.
So x = 55 is right!
Let's look back.
First: use the tangent rule.
Then: use the angle fact.
Then: group like terms.
Then: remove the + 125.
Last: check.