Search topics

Search

Triangle angle sum, special triangles and the exterior angle theorem

Start a set of 8
Step 10 of 14
Triangle with the stated angle measures marked in their sectors.84°x37°121°

The same triangle has an exterior angle of 121°. Find the adjacent interior angle x.

\[x+121^\circ=180^\circ\]
Year 9, Chapter 7, 4 difficulty levels, No calculatorBuild a worksheet

Triangle angle facts

The interior angles of a triangle have a total of 180°. Special side information can give equal or fixed angles, and an exterior angle equals the sum of its two remote interior angles

\[\angle A+\angle B+\angle C=180^\circ\]
  1. Triangle sum The three interior angles of any triangle add to 180° \(\displaystyle \angle A+\angle B+\angle C=180^\circ\)
  2. Right triangle The two acute angles in a right-angled triangle add to 90° \(\displaystyle \angle A=90^\circ,\quad \angle B+\angle C=90^\circ\)
  3. Equal base angles In an isosceles triangle, the angles opposite the equal sides are equal \(\displaystyle AB=AC\Rightarrow\angle B=\angle C\)
  4. Exterior angle An exterior angle equals the two remote interior angles added together \(\displaystyle \angle A_{\mathrm{ext}}=\angle B+\angle C\)
  5. Equilateral triangle All three sides and all three angles are equal. Each interior angle is 60° \(\displaystyle 180^\circ\div 3=60^\circ\)

Worked example

A triangular roof frame is labelled ABC. An exterior angle at A is 128° and the remote interior angle at B is 53°. Find the remote interior angle at C.

\[128^\circ=53^\circ+x\]
  1. \(\displaystyle \angle A_{\mathrm{ext}}=\angle B+\angle C\)An exterior angle equals the sum of its two remote interior angles
  2. \(\displaystyle 128^\circ=53^\circ+x\)Substitute the two displayed angle measures
  3. \(\displaystyle x=128^\circ-53^\circ\)Subtract 53° from both sides to isolate the unknown angle
  4. \(\displaystyle \angle C=75^\circ\)The remote interior angle at C is 75°

Common mistakes

Using the quadrilateral total
✕\(\displaystyle \angle A+\angle B+\angle C=360^\circ\)✓\(\displaystyle \angle A+\angle B+\angle C=180^\circ\)
A triangle has three interior angles, so their total is 180°, not the quadrilateral total of 360°
Using the apex as a base angle
✕\(\displaystyle B=C=48^\circ\)✓\(\displaystyle B=C=66^\circ\)
If AB = AC and apex A is 48°, the base angles share 180° − 48° = 132°, so each is 66°
Using the adjacent angle for an exterior angle
✕\(\displaystyle \angle A_{\mathrm{ext}}=\angle A\)✓\(\displaystyle \angle A_{\mathrm{ext}}=\angle B+\angle C\)
Use the sum of the two remote angles. The exterior and adjacent interior angles instead sum to 180°; they happen to be equal only when both are 90°

Difficulty levels

introA direct triangle angle-sum or right-triangle calculation is required
standardAn isosceles or equilateral property gives the missing angle
challengeAn exterior-angle relation or a two-step special-triangle calculation is required
examA rule or equation choice is followed by one exact angle calculation

Questions about this topic

Why do the interior angles of a triangle add to 180°?+

Draw a line through one vertex parallel to the opposite side. Alternate angles copy the other two interior angles onto that line. Together with the vertex angle they form a straight angle of 180°

How are the angles in an isosceles triangle related?+

The two angles opposite the equal sides are equal. The three angles still add to 180°, so the equal base angles can be found from the remaining total

What are the remote angles of an exterior angle?+

They are the two interior angles that are not adjacent to the exterior angle. The exterior angle equals their sum

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 7: Geometry and similarity

In this chapter

Angle facts and parallel linesTriangle angle sum, special triangles and the exterior angle theoremQuadrilateral angle sums and special-quadrilateral propertiesPolygon angle sums and regular polygonsCongruent triangle testsProve congruence and use it to prove a further resultEnlargement transformation and scale factor7G Enlargement and scale: Similar figures and scale factorsTests for similar triangles7H Similar triangle tests: Similar figures and scale factorsApplied similarity and scaleSimilar solids and scaling volumeDesign, test and refine geometrical algorithms