Angle facts
Use the position of the angles to choose the relation before substituting their measures. A complete turn is 360 degrees, while a straight line is 180 degrees
\[a+b+c+x=360^\circ\]
- A straight line Adjacent angles on a straight line have a total of 180 degrees \(\displaystyle 43^\circ+137^\circ=180^\circ\)
- Angles at a point All angles around one point make one complete turn of 360 degrees \(\displaystyle a+b+c+x=360^\circ\)
- Vertically opposite angles Opposite angles formed by two crossing lines are equal \(\displaystyle x=58^\circ\)
- Parallel-line pairs Corresponding and alternate angles are equal, and cointerior angles total 180 degrees, when the two lines are parallel \(\displaystyle a=b,\quad c+d=180^\circ\)
- Acute and right An acute angle is greater than 0 degrees and less than 90 degrees. A right angle is exactly 90 degrees
- Obtuse and straight An obtuse angle is greater than 90 degrees and less than 180 degrees. A straight angle is exactly 180 degrees
- Reflex and full turn A reflex angle is greater than 180 degrees and less than 360 degrees. One full turn, or revolution, is exactly 360 degrees
- Testing parallelism Equal corresponding or alternate angles, or a cointerior sum of 180 degrees, proves parallelism. With exact measures, a failed relation rules it out
Worked example
Four angles around one point are 52, 86, 71 and x degrees. Find x
\[a+b+c+x=360^\circ\]
- \(\displaystyle 52+86+71=209\)Add the three known angle measures, in degrees
- \(\displaystyle 209+x=360\)All four sectors make one complete turn
- \(\displaystyle x=360-209=151\)The unknown angle is 151 degrees
Common mistakes
Using a straight-line total at a point
✕\(\displaystyle a+b+c+x=180^\circ\)✓\(\displaystyle a+b+c+x=360^\circ\)
All sectors around a point make a complete turn. Use 180 degrees only for a straight angleTreating vertical angles as supplementary
✕\(\displaystyle x+58^\circ=180^\circ\)✓\(\displaystyle x=58^\circ\)
Vertically opposite angles are equal. Adjacent angles on a straight line total 180 degrees. At a perpendicular crossing both statements holdUsing the complementary total
✕\(\displaystyle x=90^\circ-37^\circ=53^\circ\)✓\(\displaystyle x=180^\circ-37^\circ=143^\circ\)
Complementary angles total 90 degrees, but a supplementary straight-line pair totals 180 degreesDifficulty levels
introClassify an angle or use a direct vertical, complementary or supplementary fact
standardIdentify a corresponding, alternate or cointerior pair and follow a two- or three-angle chain
challengeTest a converse relation to decide whether described lines are parallel
examChoose the angle reason at each stage before finding an exact final angle
Questions about this topic
How are corresponding and alternate angles different?+
Corresponding angles occupy matching positions at the two crossings. Alternate angles are between the parallel lines on opposite sides of the transversal. Both pairs are equal when the lines are parallel
When do angles add to 90 degrees?+
Angles add to 90 degrees when they are complementary. This is different from supplementary angles, which add to 180 degrees
How can a pair of angles prove that lines are parallel?+
Use a converse: equal corresponding angles, equal alternate angles or cointerior angles with a total of 180 degrees proves that the lines are parallel
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