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Triangle angle sum, special triangles and the exterior angle theorem

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Triangle with the stated angle measures marked in their sectors.xx44°

An isosceles roof frame has two equal sloping sides. Its apex angle is 44°. Find each equal base angle x

Year 10, 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

  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° \(\displaystyle \begin{gathered}\angle A=90^\circ\\\angle B+\angle C=90^\circ\end{gathered}\)
  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° \(\displaystyle 180^\circ\div 3=60^\circ\)
  6. A general triangle-sum proof Draw a parallel to one side through the opposite vertex. Alternate angles copy the other two angles beside the vertex angle, forming a straight angle
  7. Deriving the exterior-angle theorem The triangle sum and straight-angle sum have the same total. Subtract the shared adjacent angle \(\displaystyle \begin{gathered}a+b+c=180^\circ\\b+e=180^\circ\\e=a+c\end{gathered}\)

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°
Confusing evidence with proof
✕\(\displaystyle \text{Measure one drawing}\)✓\(\displaystyle \text{Use general angle facts}\)
Measurements suggest a relationship. A proof explains why it holds for all triangles satisfying the assumptions

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
examChoose a triangle classification, rule or equation, then calculate an exact angle

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 a triangle have two right or two obtuse angles?+

No. Two right angles leave zero for the third angle, and two obtuse angles exceed 180 degrees before the third angle is added. A non-degenerate triangle has three positive interior angles

Why is an exterior angle larger than either remote interior angle?+

It is their sum, and both remote angles are positive

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 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 testsTests for similar triangles7F Similar triangle tests: Similar figures and scale factors7G Applied triangle similarity: Applied similarity and scaleSimilar solids and scaling volume7H Similar solids and map scale: Similar figures and scale factors7H Similar solids and map scale: Applied similarity and scaleInterpret practical networks