Worked examples
The question
Question: find x.
ABCD is a cyclic quadrilateral.
The opposite angles at A and C are 110° and x.
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: 110 + x = 180.
110 + x = 180
Careful!
Opposite angles are equal in a parallelogram.
In a cyclic quadrilateral they are not equal.
They add up to 180°.
Careful: not 110 = x ✗
2 Remove the 110
We want x by itself.
The 110 is in the way.
To remove 110, we take away 110.
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.
110 + x − 110 = 180 − 110
110 and − 110 make zero.
Cross them out.
180 − 110 = 70.
180 − 110 = 70
So x = 70!
x = 70
3 Check
Let's check: put x = 70 back and add the opposite angles.
110 + 70 = 180
They add up to 180°.
So x = 70 is right!
Let's look back.
First: use the circle rule.
Then: remove the 110.
Last: check.
▶ Watch it solved
The question
Question: find x.
ABCD is a cyclic quadrilateral.
The opposite angles at A and C are 2x and 3x.
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 + 3x = 180.
2x + 3x = 180
Careful!
Opposite angles are equal in a parallelogram.
In a cyclic quadrilateral they are not equal.
They add up to 180°.
Careful: not 2x = 3x ✗
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.
2x + 3x = 5x.
2x + 3x = 5x
Now we have 5x = 180.
5x = 180
3 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.
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.
5x5 = 1805
5 divided by 5 is 1.
So just x is left.
180 divided by 5 is 36.
1805 = 36
So x = 36!
x = 36
4 Check
Let's check: put x = 36 in each angle.
2x = 2 × 36 = 72
3x = 3 × 36 = 108
Now add the opposite angles.
72 + 108 = 180
They add up to 180°.
So x = 36 is right!
Let's look back.
First: use the circle rule.
Then: group like terms.
Then: remove the 5.
Last: check.
▶ Watch it solved
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.
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.
y + 105 − 105 = 180 − 105
+ 105 and − 105 make zero.
Cross them out.
180 − 105 = 75.
180 − 105 = 75
So y = 75!
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.
x + 75 = 180
Careful!
The angle at A is opposite C.
Opposite angles add up to 180°.
They are not equal.
Careful: not x = 75 ✗
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.
x + 75 − 75 = 180 − 75
+ 75 and − 75 make zero.
Cross them out.
180 − 75 = 105.
180 − 75 = 105
So x = 105!
x = 105
5 Check
Let's check: put x = 105 back and add the opposite angles.
105 + 75 = 180
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.
▶ Watch it solved
