SPI Mathematics · Lines, Angles and Polygons

Intersecting Lines, Parallel Lines and Polygons

Where two lines cross, opposite angles are equal and neighbours add to 180 degrees; a transversal across parallel lines makes equal alternate and corresponding angles; the exterior angles of any polygon add to 360 degrees.

All Lines, Angles and Polygons notes

Why this matters

Five questions in the bank, two EASY and three MODERATE: angles round a crossing point, a bent path between parallel lines, and the sides of a regular polygon from one exterior angle. Each needs one rule applied twice, so name the rule at every step.

Concept 1 of 3: Angles where two lines cross

Two crossing lines make four angles round one point. Neighbouring angles sit on a straight line, so they add to 180 degrees, and the angles facing each other across the point come out equal.

Definition

Neighbouring angles on a straight line form a linear pair and add to $180^\circ$ .
Vertically opposite angles, facing each other across the crossing point, are equal.
All the angles round a point add to $360^\circ$ .
So one angle gives all four: the opposite one is equal and the two others are $180^\circ$ minus it.

Angles at a crossing

\text{linear pair: } a + b = 180^\circ \qquad \text{vertically opposite angles are equal} \qquad \text{round a point: } 360^\circ

Worked example

Lines PQ and RS cross at O, with

\angle POR = 3x

and

\angle ROQ = 2x

. Find all four angles at O.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Lines, Angles and Polygons · Intersecting Lines, Parallel Lines and Polygons

Two straight lines AB and CD intersect one another at the point O. If

\angle

AOC +

\angle

COB +

\angle

BOD =

274^\circ

, then

\angle

AOD =

Making vertically opposite angles add to 180

Vertically opposite angles are equal. It is the neighbouring angles, the linear pair, that add to

180^\circ

Using 180 for the angles round a point

All the angles round a point add to

360^\circ

. Only the angles on one side of a straight line add to

180^\circ

Drill 1 more on angles where two lines cross

Concept 2 of 3: Parallel lines cut by a transversal

A line crossing two parallel lines meets both at the same tilt, so the angles it makes at one line repeat at the other. That gives equal pairs and pairs adding to 180 degrees.

Definition

A transversal is a line that cuts two or more lines.
If the lines are parallel: alternate angles (a Z shape) are equal, corresponding angles (an F shape) are equal, and co-interior angles (a C shape, same side, between the lines) add to $180^\circ$ .
It works the other way too: if alternate angles are equal, the lines are parallel.
For a bent path between two parallel lines, draw a third line through the bend, parallel to both. The angle at the bend is then the sum of the two alternate angles.

Parallel lines

\text{alternate} = \text{alternate} \qquad \text{corresponding} = \text{corresponding} \qquad \text{co-interior angles add to } 180^\circ

Worked example

A transversal cuts two parallel lines, and one of the angles it makes with the first line is

65^\circ

. Find the alternate angle and the co-interior angle at the second line.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Lines, Angles and Polygons · Intersecting Lines, Parallel Lines and Polygons

In the figure, AB || CD || PQ and

\angle BKP = 40^\circ

and

\angle DLP = 30^\circ

, then the value of x is:

Making co-interior angles equal

Co-interior angles, on the same side and between the parallel lines, add to

180^\circ

. Alternate and corresponding angles are the equal ones.

Using the rules on lines that are not parallel

Alternate angles are equal only when the lines are parallel. If a question asks when two lines are parallel, set the alternate angles equal and solve.

Drill 1 more on parallel lines cut by a transversal

Concept 3 of 3: Angles of a triangle and a polygon

Walk round any polygon and you turn once all the way round, so the exterior angles always add to 360 degrees. In a regular polygon all of them are equal, so one exterior angle tells you the number of sides.

Definition

The angles of a triangle add to $180^\circ$ .
The interior angles of a polygon with $n$ sides add to $(n - 2) \times 180^\circ$ .
The exterior angles of any polygon add to $360^\circ$ .
In a regular polygon: each exterior angle is $\frac{360^\circ}{n}$ , and each interior angle is $180^\circ$ minus it.

Polygon angles

\text{interior sum} = (n - 2) \times 180^\circ \qquad \text{exterior angle of a regular polygon} = \frac{360^\circ}{n}

nnumber of sides

Worked example

Each exterior angle of a regular polygon is

40^\circ

. How many sides does it have, and what is each interior angle?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Lines, Angles and Polygons · Intersecting Lines, Parallel Lines and Polygons

Find the number of sides of a regular polygon whose each exterior angle has a measure of

45^\circ

Dividing 360 by the interior angle

The number of sides is 360 divided by the EXTERIOR angle. If the interior angle is given, subtract it from

180^\circ

first.

Using 180 for the exterior angles

The exterior angles of every polygon add to

360^\circ

, whatever the number of sides. It is the triangle's interior angles that add to

180^\circ

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Angles where two lines cross
Angles at a crossing
$\text{linear pair: } a + b = 180^\circ \qquad \text{vertically opposite angles are equal} \qquad \text{round a point: } 360^\circ$
Parallel lines cut by a transversal
Parallel lines
$\text{alternate} = \text{alternate} \qquad \text{corresponding} = \text{corresponding} \qquad \text{co-interior angles add to } 180^\circ$
Angles of a triangle and a polygon
Polygon angles
$\text{interior sum} = (n - 2) \times 180^\circ \qquad \text{exterior angle of a regular polygon} = \frac{360^\circ}{n}$

Watch out for (6)

Making vertically opposite angles add to 180
→ Angles where two lines cross
Using 180 for the angles round a point
→ Angles where two lines cross
Making co-interior angles equal
→ Parallel lines cut by a transversal
Using the rules on lines that are not parallel
→ Parallel lines cut by a transversal
Dividing 360 by the interior angle
→ Angles of a triangle and a polygon
Using 180 for the exterior angles
→ Angles of a triangle and a polygon

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