FG3–FG4, HG3–HG4 - Angles in triangles and polygons

FG3–FG4, HG3–HG4 - Angles in triangles and polygons

Why does adding a side to a polygon add 180∘180^\circ to its interior angle total? Derive triangle and polygon angle sums using FG3 and HG3, then use the special-triangle properties in FG4 and HG4 to find missing angles. Connect the angles of regular polygons to one full turn.

Triangle sum

Could three angles of 70∘70^\circ, 65∘65^\circ and 55∘55^\circ belong to a triangle? Their total, rather than the appearance of a sketch, decides.

Draw a line through vertex AA of triangle ABCABC, parallel to BCBC. The angles between that line and ABAB, and between that line and ACAC, equal the angles at BB and CC by alternate angles. Together with the angle at AA, they fill a straight angle.

[DIAGRAM: asset_name: Deriving the triangle angle sum; asset_slug: triangle-angle-derivation; caption: A line parallel to the base brings the three triangle angles together on a straight line.; description: A scalene triangle ABC has a horizontal base BC and a horizontal line through A marked parallel to BC. Angles at B and C are labelled b and c, the angle at A is a, and the corresponding alternate sectors at A are labelled b and c.]
A line parallel to the base brings the three triangle angles together on a straight line.

Triangle angle sum

The three interior angles of a triangle add to 180∘180^\circ:

a+b+c=180∘a+b+c=180^\circ

This applies to every triangle in a flat plane. Not in the formula booklet: learn it.

An isosceles triangle has two equal sides and equal angles opposite them. An equilateral triangle has all three sides equal, so all three angles are equal: 180∘÷3=60∘180^\circ\div3=60^\circ. In a right-angled triangle, the other two angles sum to 90∘90^\circ. Equal sides need not be drawn upright; find the angle opposite each marked side.

Worked example 1: an isosceles triangle

Question. In triangle ABCABC, AB=ACAB=AC and angle BAC=38∘BAC=38^\circ. Work out the two base angles.

Plan. Subtract the vertex angle, then share the remaining total equally.

∠ABC+∠BCA=180∘−38∘=142∘triangle sum∠ABC=∠BCA=142∘÷2=71∘equal base angles\begin{aligned} \angle ABC+\angle BCA&=180^\circ-38^\circ=142^\circ &&\text{triangle sum}\\ \angle ABC=\angle BCA&=142^\circ\div2=71^\circ &&\text{equal base angles} \end{aligned}

Answer. Both base angles are 71∘71^\circ.

Check. 38+71+71=18038+71+71=180, and the angles opposite the equal sides match.

Change which angle is known: the equal pair is now the starting information.

Worked example 2: a known base angle

Question. In triangle PQRPQR, PQ=QRPQ=QR and angle QPR=47∘QPR=47^\circ. Work out angle PQRPQR. The side PQPQ is then extended past QQ to SS; find angle SQRSQR.

Plan. Identify the equal base angles, use the triangle sum, then a straight line.

∠PRQ=47∘opposite equal sides∠PQR=180∘−2(47∘)=86∘triangle sum∠SQR=180∘−86∘=94∘straight line\begin{aligned} \angle PRQ&=47^\circ &&\text{opposite equal sides}\\ \angle PQR&=180^\circ-2(47^\circ)=86^\circ &&\text{triangle sum}\\ \angle SQR&=180^\circ-86^\circ=94^\circ &&\text{straight line} \end{aligned}

Answer. ∠PQR=86∘\angle PQR=86^\circ, ∠SQR=94∘\angle SQR=94^\circ.

Check. The exterior angle 94∘94^\circ equals the two remote interior angles 47∘+47∘47^\circ+47^\circ. In any triangle this follows by subtracting its third angle from 180∘180^\circ.

Use the equality markings before deciding which angle to double.

Watch out

An isosceles triangle does not have three equal angles unless it is equilateral. If the vertex angle is 38∘38^\circ, dividing 180∘180^\circ into three equal pieces discards the given information.

Find a vertex angle and its adjacent exterior angle.

Polygon sums

Split a polygon into triangles to turn a many-sided problem into a familiar one.

A quadrilateral splits into two triangles along a diagonal, giving 2×180∘=360∘2\times180^\circ=360^\circ. A pentagon splits into three triangles, giving 540∘540^\circ. Each extra vertex adds one more triangle.

[DIAGRAM: asset_name: Triangulating a polygon; asset_slug: polygon-triangulation; caption: A six-sided polygon splits into four triangles without adding an interior vertex.; description: An irregular convex hexagon ABCDEF has diagonals from A to C, D and E, creating four lightly shaded triangles. All six vertices are labelled, and text states six sides, four triangles.]
A six-sided polygon splits into four triangles without adding an interior vertex.

Interior angle total

For a simple polygon with nn sides,

interior angle total=(n−2)×180∘\text{interior angle total}=(n-2)\times180^\circ

There are n−2n-2 triangles. The formula also holds for a concave polygon, including any reflex interior angles, if it is split into triangles without overlap. Not in the formula booklet: learn it.

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account