G3 - Angle facts, parallel lines and polygons

Angles at a point and on a line

Could a regular polygon have an interior angle of $156^\circ$? You will find its number of sides by starting with the simplest angle facts.

A complete turn is $360^\circ$; a straight angle is half of this, $180^\circ$. An acute angle lies between $0^\circ$ and $90^\circ$, an obtuse angle between $90^\circ$ and $180^\circ$, and a reflex angle between $180^\circ$ and $360^\circ$. The reflex turn between two rays is $360^\circ$ minus their smaller angle.

Check that the named angles fill exactly the required turn: no gap and no overlap. For example, angles $94^\circ$, $137^\circ$ and $y$ filling a point give $y=360^\circ-94^\circ-137^\circ=129^\circ$.

When two straight lines cross, vertically opposite angles are equal. If adjacent pairs satisfy $a+b=180^\circ$ and $b+c=180^\circ$, subtracting the shared $b$ gives $a=c$. This result is not on the formula sheet: derive it from the straight-line angle sum. “Vertically” refers to a common vertex, not the page's orientation.

Change only the way the unknown angle is labelled: it is now $2x$ degrees. First find the angle, then find the variable.

The same care applies to more involved labels. Adjacent angles $3x+10$ and $2x+20$ degrees give $5x+30=180$, so $x=30$ and the actual angles are $100^\circ$ and $80^\circ$. At an intersection with one angle $47^\circ$, its opposite angle is $47^\circ$ and its neighbours are $133^\circ$.

Change only the coefficient of the unknown label for your turn.

Angles on parallel lines

A transversal crosses two lines; when those lines are parallel, it links angles at the two crossings.

Corresponding angles occupy matching positions, such as above each parallel line and to the right of the transversal. Alternate interior angles lie between the parallel lines on opposite sides of the transversal. Both pairs are equal on parallel lines. Alternate exterior angles, outside the parallels on opposite sides of the transversal, are equal too.

Co-interior angles are between the parallels on the same side of the transversal. They add to $180^\circ$: transfer one by the alternate-angle rule, then use a straight angle. F and Z shapes can help you spot positions, but the parallel condition and the angle positions supply the reason. These parallel-line rules are not on the formula sheet: learn them.

[DIAGRAM: asset_name: Parallel-line angle positions; asset_slug: 1ma1_m46_parallel_angles; caption: Corresponding and alternate angles are equal; co-interior angles add to 180 degrees.; description: Three panels show a sloping transversal crossing marked parallel horizontal lines. Coloured arcs identify corresponding angles, alternate interior angles and a co-interior pair. A 60-degree example gives equal 60-degree pairs and a 60-degree plus 120-degree co-interior pair.]

For instance, a corresponding angle to $63^\circ$ is $63^\circ$, and its neighbour is $180^\circ-63^\circ=117^\circ$. State the particular rule, not just “parallel lines”.

Change only the label of the second angle to $3x$ degrees. The geometrical reason remains the same.

The rules work backwards too. Equal corresponding angles, equal alternate angles, or co-interior angles totalling $180^\circ$ establish that the two lines are parallel. Thus co-interior angles of $72^\circ$ and $108^\circ$ prove parallelism even if no parallel arrows were supplied. The angle positions relative to the same transversal are essential evidence.

In a reasoned solution, identify the angles clearly using three letters when a vertex has several angles. This makes each stage unambiguous. The next question changes only the coefficient in the unknown label.

Deriving the triangle angle sum

You can prove the triangle angle sum with one extra parallel line, so you do not need to measure thousands of triangles.

In triangle $ABC$, draw a line through $A$ parallel to $BC$. Alternate angles transfer angle $B$ to one side of $A$, and angle $C$ to the other. These transferred angles and the original apex angle fill a straight angle at $A$.