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 105°.
The opposite angle, at A, is x.
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 105° add up to 180°.
So y + 105 = 180.
2 Remove the + 105
We want y by itself.
The + 105 is in the way.
To remove + 105, we take away 105.
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.
+ 105 and − 105 make zero.
Cross them out.
180 − 105 = 75.
So y = 75!
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 x + 75 = 180.
Careful!
The angle at A is opposite C.
Opposite angles add up to 180°.
They are not equal.
4 Remove the + 75
We want x by itself.
The + 75 is in the way.
To remove + 75, we take away 75.
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.
+ 75 and − 75 make zero.
Cross them out.
180 − 75 = 105.
So x = 105!
5 Check
Let's check: put x = 105 back and add the opposite angles.
They add up to 180°.
And 105° is the same as the angle outside: 105°.
So x = 105 is right!
The angle outside always equals the opposite angle inside.
Let's look back.
First: use angles on a line.
Then: remove the + 105.
Then: use the circle rule.
Then: remove the + 75.
Last: check.