Optional extension: Circle theorems, tangent angles and regular polygon exterior angles
Example 1
ABCD is a cyclic quadrilateral. Opposite angles A and C are (2x+31)∘ and (3x+59)∘. Find angle C.
Cyclic quadrilateralDiagram not to scaleShow worked answer
Answer: 113
- Opposite angles sum to 180∘.
- Thus (2x+31)+(3x+59)=180, or 5x+90=180.
- This gives x=18.
- Substituting into angle C gives 113∘.
Example 2
A regular polygon has 8 sides. The marked exterior angle is (x+40)∘. Find x.
Regular polygonDiagram not to scaleShow worked answer
Answer: 5
- Each exterior angle is 8360=45∘.
- Hence x+40=45, giving x=5.
Example 3
ABCD is a cyclic quadrilateral. Opposite angles A and C are (x+34)∘ and (2x+71)∘. Find angle A.
Cyclic quadrilateralDiagram not to scaleShow worked answer
Answer: 59
- Opposite angles sum to 180∘.
- Thus (x+34)+(2x+71)=180, or 3x+105=180.
- This gives x=25.
- Substituting into angle A gives 59∘.