Year 7AC9M7M05

Angle Sums of Triangles and Polygons

Demonstrate that the interior angle sum of a triangle in the plane is 180 degrees and apply this to determine the interior angle sum of other shapes and the size of unknown angles.

Every polygon splits into triangles, and every triangle holds 180 degrees. One fact, extended, gives you the angle sum of any shape.

Method

Triangle

\text{angle sum} = 180^\circ

True for every triangle, whatever its shape.

Any polygon

S = (n-2) \times 180^\circ

$n$ is the number of sides. Split the shape into triangles to see why.

Always give a reason

Write the rule you used beside every answer.

Worked example

e.g.

\text{Find the interior angle sum of a hexagon, then one angle of a regular hexagon.}

Count the sides. $n = 6$
Split it into triangles from one corner. There are always two fewer. $6 - 2 = 4 \text{ triangles}$
Each triangle holds $180^\circ$ . $S = 4 \times 180^\circ = 720^\circ$
Regular means all angles equal, so divide by the number of angles. $720 \div 6 = 120^\circ$
State the answer with its reason. $120^\circ \ [\text{angle sum of a hexagon}]$

Practice

Give a reason for every answer.

\text{Two angles of a triangle are } 55^\circ \text{ and } 68^\circ. \text{ Find the third.}

Answer

57^\circ \ [\text{angle sum of a triangle}]

\text{Angle sum of a pentagon}

Answer

540^\circ

\text{One angle of a regular octagon}

Answer

1080 \div 8 = 135^\circ

\text{A quadrilateral has angles } 90,90,100,x. \text{ Find } x.

Answer

80^\circ

\text{Why is the angle sum } (n-2)\times180?

Answer

\text{A polygon splits into } n-2 \text{ triangles.}