Step by step
The question
Question: find x.
ABCD is a cyclic quadrilateral.
The opposite angles at A and C are 5x + 10 and 4x − 1.
1 Use the circle rule
All four corners, A, B, C and D, are on the circle.
This is called a cyclic quadrilateral.
Join B and D to the center, O.
The angle at A stands on the arc through C.
The circle rule: the angle at the center on that arc is double: 2 times A.
The angle at C stands on the other arc.
Its angle at the center is 2 times C.
Together, the two angles at the center go all the way round: 360°.
Half of that: A and C add up to 180°.
Opposite angles always add up to 180°.
So we add them and make it 180: 5x + 10 + 4x − 1 = 180.
Careful!
Opposite angles are equal in a parallelogram.
In a cyclic quadrilateral they are not equal.
They add up to 180°.
2 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.
5x + 4x = 9x.
10 − 1 = 9.
Now we have 9x + 9 = 180.
We want x by itself.
Two things are stuck to x: times 9 and + 9.
The + 9 is on the outside, so it comes off first.
Then the times 9.
3 Remove the + 9
To remove + 9, we take away 9.
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.
+ 9 and − 9 make zero.
Cross them out.
180 − 9 = 171.
Now we have 9x = 171.
4 Remove the 9
9x means 9 times x.
We want just x.
To remove times 9, we divide by 9.
We do it on both sides, so they stay equal.
9 divided by 9 is 1.
So just x is left.
171 divided by 9 is 19.
So x = 19!
5 Check
Let's check: put x = 19 in each angle.
Now add the opposite angles.
They add up to 180°.
So x = 19 is right!
Let's look back.
First: use the circle rule.
Then: group like terms.
Then: remove the + 9.
Then: remove the 9.
Last: check.