Interior and exterior angles

Geometry and measures

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Write the sum of the interior angles of a triangle.

2 marks
Write the sum of the interior angles of a quadrilateral.

2 marks
A regular hexagon has six sides. Find one exterior angle and one interior angle.

3 marks
Find the sum of the interior angles of a pentagon.

3 marks
A regular polygon has exterior angle $24°$. How many sides does it have, and what is each interior angle?

4 marks
An isosceles triangle has two angles of $72°$. Find the third angle. Then show that an exterior angle at the third vertex equals the sum of the two remote interior angles.

4 marks
Four interior angles of a pentagon are $100°$, $110°$, $120°$ and $130°$. Find the fifth.

5 marks
A regular polygon has interior angle $150°$. Find the number of sides and the sum of the interior angles. Show that the exterior angles sum to $360°$.

6 marks

Mark scheme

Answer. $180°$
- The angles of a triangle sum to $180°$.
Answer. $360°$
- A quadrilateral splits into two triangles.
- $2 \times 180 = 360°$
Answer. Exterior $60°$, interior $120°$
- Exterior $= \dfrac{360}{6} = 60°$.
- Interior $= 180 - 60 = 120°$.
Answer. $540°$
- $(5 - 2) \times 180$
- $540°$
Answer. $15$ sides, interior angle $156°$
- $360 \div 24 = 15$ sides.
- Interior angle $= 180 - 24 = 156°$.
Answer. The third angle is $36°$. Its exterior angle is $144°$, which equals $72° + 72°$.
- $180 - 72 - 72 = 36°$
- Exterior angle $= 180 - 36 = 144°$.
- $72 + 72 = 144$
Answer. $80°$
- The interior angles of a pentagon sum to $540°$.
- $100 + 110 + 120 + 130 = 460$
- Fifth angle $= 540 - 460 = 80°$.
Answer. $12$ sides. The interior sum is $1800°$. Twelve exterior angles of $30°$ sum to $360°$.
- Exterior angle $= 180 - 150 = 30°$.
- Number of sides $= 360 \div 30 = 12$.
- Interior sum $= (12 - 2) \times 180 = 1800°$.
- $12 \times 30 = 360°$.