Step by step
The question
Question: find x.
The angles of a quadrilateral are x + 100, 2x + 120, 80° and 90°.
1 Find the angle sum
Pick one corner.
Draw lines from it to the other corners.
They cut the quadrilateral into 2 triangles.
There are always 2 fewer triangles than sides: 4 − 2 = 2.
Each triangle has 180°.
2 times 180 is 360.
2 Use the angle fact
The angles of the quadrilateral add up to 360°.
So we add them all and make it 360: x + 100 + 2x + 120 + 80 + 90 = 360.
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.
100 + 120 + 80 + 90 = 390.
Now we have 3x + 390 = 360.
We want x by itself.
Two things are stuck to x: times 3 and + 390.
The + 390 is on the outside, so it comes off first.
Then the times 3.
4 Remove the + 390
To remove + 390, we take away 390.
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.
+ 390 and − 390 make zero.
Cross them out.
360 − 390 = −30.
Now we have 3x = −30.
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.
−30 divided by 3 is −10.
So x = −10!
6 Check
Let's check: put x = −10 in each angle.
Now add all the angles.
They add up to 360°.
So x = −10 is right!
Let's look back.
First: find the angle sum.
Then: use the angle fact.
Then: group like terms.
Then: remove the + 390.
Then: remove the 3.
Last: check.