Step by step
The question
Question: find x.
AB is a diameter.
The angles at A and B are x + 10 and 2x + 20.
1 Use the circle rule
AB goes through the center O.
It is a diameter.
It cuts the circle in half: two semicircles.
At the center, A and B make a straight line: 180°.
The angle at C, on the edge, is half of it: 90°.
A right angle.
2 Use the angle fact
The three angles in a triangle always add up to 180°.
So we add them all and make it 180: x + 10 + 2x + 20 + 90 = 180.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
x + 2x = 3x.
10 + 20 + 90 = 120.
Now we have 3x + 120 = 180.
We want x by itself.
Two things are stuck to x: times 3 and + 120.
The + 120 is on the outside, so it comes off first.
Then the times 3.
4 Remove the + 120
To remove + 120, we take away 120.
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.
+ 120 and − 120 make zero.
Cross them out.
180 − 120 = 60.
Now we have 3x = 60.
5 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
60 divided by 3 is 20.
So x = 20!
6 Check
Let's check: put x = 20 in each angle.
Now add all the angles.
They add up to 180°.
So x = 20 is right!
Let's look back.
First: use the circle rule.
Then: use the angle fact.
Then: group like terms.
Then: remove the + 120.
Then: remove the 3.
Last: check.