Step by step
The question
Question: find x.
ABCD is a cyclic quadrilateral.
The opposite angles at A and C are 2x + 10 and 3x − 5.
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: 2x + 10 + 3x − 5 = 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.
2x + 3x = 5x.
10 − 5 = 5.
Now we have 5x + 5 = 180.
We want x by itself.
Two things are stuck to x: times 5 and + 5.
The + 5 is on the outside, so it comes off first.
Then the times 5.
3 Remove the + 5
To remove + 5, we take away 5.
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.
+ 5 and − 5 make zero.
Cross them out.
180 − 5 = 175.
Now we have 5x = 175.
4 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
175 divided by 5 is 35.
So x = 35!
5 Check
Let's check: put x = 35 in each angle.
Now add the opposite angles.
They add up to 180°.
So x = 35 is right!
Let's look back.
First: use the circle rule.
Then: group like terms.
Then: remove the + 5.
Then: remove the 5.
Last: check.