Step by step
The question
Question: find x.
ABCD is a cyclic quadrilateral.
The side DC is carried on past C.
The angle outside at C is 115°.
The opposite angle, at A, is 2x + 15.
1 Use angles on a line
The dashed line carries on the side DC.
It is a straight line.
So the angle inside at C, call it y, and 115° add up to 180°.
So y + 115 = 180.
2 Remove the + 115
We want y by itself.
The + 115 is in the way.
To remove + 115, we take away 115.
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.
+ 115 and − 115 make zero.
Cross them out.
180 − 115 = 65.
So y = 65!
3 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 2x + 15 + 65 = 180.
Careful!
The angle at A is opposite C.
Opposite angles add up to 180°.
They are not equal.
4 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.
15 + 65 = 80.
Now we have 2x + 80 = 180.
We want x by itself.
Two things are stuck to x: times 2 and + 80.
The + 80 is on the outside, so it comes off first.
Then the times 2.
5 Remove the + 80
To remove + 80, we take away 80.
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.
+ 80 and − 80 make zero.
Cross them out.
180 − 80 = 100.
Now we have 2x = 100.
6 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
100 divided by 2 is 50.
So x = 50!
7 Check
Let's check: put x = 50 in the angle at A.
Now add the opposite angles.
They add up to 180°.
And 115° is the same as the angle outside: 115°.
So x = 50 is right!
The angle outside always equals the opposite angle inside.
Let's look back.
First: use angles on a line.
Then: remove the + 115.
Then: use the circle rule.
Then: group like terms.
Then: remove the + 80.
Then: remove the 2.
Last: check.