8.03d, 8.04a–8.04c - Angles and properties of polygons

Triangle properties and angles

A regular octagon has eight equal sides and eight equal angles. How large is each angle? Rather than measuring a particular drawing, start with a triangle: it is the building block for the calculation.

[DIAGRAM: asset_name: Deriving the triangle angle sum; asset_slug: triangle-angle-sum; caption: A parallel line through A puts all three triangle angles into one straight angle.; description: Triangle ABC has A above the horizontal base BC. A line through A is marked parallel to BC. The triangle's interior angles are a at A, b at B and c at C. The angles on either side of a beneath the upper line are labelled b and c by alternate-angle equality. A second panel has isosceles triangle PQR with PQ equal to PR, vertex angle v at P and equal base angles u. A third panel shows triangle XYZ right-angled at Y, with angle X 32 degrees and angle Z 58 degrees.]

Through A draw a line parallel to BC. The angle next to AB on that line equals angle B, and the angle next to AC equals angle C, by alternate angles. Along the straight line at A, those two angles and angle A sum to $180^\circ$. Therefore the three interior angles of a triangle sum to $180^\circ$.

An isosceles triangle has at least two equal sides. The angles opposite those sides are equal: in PQR, $PQ=PR$ gives $\angle PQR=\angle PRQ$. One way to see the reason is to fold along the line from P to the midpoint of QR: the two halves match, so the base angles match. We will justify this precisely using congruent triangles in a later lesson. Conversely, two equal angles have equal opposite sides.

Change only the vertex angle. A larger angle at the top leaves a smaller total for the base pair.

In an equilateral triangle, all three sides are equal, so all three opposite angles are equal. Their total is $180^\circ$, giving $60^\circ$ each. A right-angled triangle has one angle of $90^\circ$, so its other two angles total $90^\circ$. In the right-angled triangle XYZ shown, angle X is $32^\circ$, so angle Z is $90^\circ-32^\circ=58^\circ$. A right-angled isosceles triangle therefore has two $45^\circ$ angles.

You can also identify an isosceles triangle from two equal angles, even if equal sides are not marked. State that the sides opposite equal angles are equal; do not say “it looks symmetrical”.

Use fresh data in the same labelled isosceles arrangement.

Interior and exterior angle sums

A diagonal divides a polygon into triangles. That lets a rule for three sides explain the angle total for many sides.

[DIAGRAM: asset_name: Triangles and turns in a pentagon; asset_slug: polygon-triangles-and-turns; caption: The pentagon contains three triangles. Walking its boundary makes one complete turn.; description: The left panel shows convex pentagon ABCDE with diagonals AC and AD, dividing it into three lightly shaded triangles. The right shows the same pentagon with each side extended in the direction of travel and a marked exterior turn at each vertex. One interior angle i and its adjacent exterior angle e are labelled.]

For a convex polygon, each interior angle is less than $180^\circ$, and diagonals from one vertex lie inside the shape. A pentagon splits into three triangles, a hexagon into four, and an $n$-sided polygon into $n-2$. The triangle angles together make exactly the polygon's interior angles.

An exterior angle here is the turn from the straight continuation of the incoming side to the outgoing side as you travel around a convex polygon. It is adjacent to the interior angle, so

Do not use the reflex angle outside the shape instead. Walking once around the boundary returns you to your original direction after one complete turn. Hence the exterior angles total $360^\circ$, even if the side lengths and angles differ. Algebra confirms it: $n\times180^\circ-(n-2)\times180^\circ=360^\circ$. For a concave boundary, turns must be treated with signs; adding only positive smaller angles is not the same rule.

Change only the number of sides from six to seven. One extra side adds one triangle to the triangulation.

The exterior total stays fixed while the interior total grows. That difference will be useful when a question gives you angles but asks for the number of sides.

Apply the relationship between the two totals.

Angles of regular polygons

In a regular polygon all the angles are equal, so divide the total by the number of angles. In an irregular polygon that division gives a mean, not necessarily any actual angle.

[DIAGRAM: asset_name: Regular octagon angles; asset_slug: regular-octagon-angles; caption: Eight equal turns divide 360 degrees equally. Each interior angle completes a straight angle with its turn.; description: A regular octagon has one side extended at vertex A. The interior angle i is shown inside, and its adjacent exterior turn e outside. Matching side ticks identify equal lengths, and the caption states that all eight interior angles are equal.]

Change only the number of sides to twelve. More sides mean smaller equal turns and larger interior angles.

You can reverse the process when the angle is given instead of the number of sides. First recover the exterior turn; then count how many such turns fit one full turn.