Lines, Angles and Polygons — SPI Mathematics Formulas and Traps

Lines, Rays and Angle Pairs

Complementary and supplementary angles

Complement and supplement

\text{complement of } x = 90^\circ - x \qquad \text{supplement of } x = 180^\circ - x

Lines, rays, segments and kinds of angle

Term	Meaning
Line	A straight path going on endlessly in both directions, with no endpoints
Line segment	A part of a line with two endpoints
Ray	A part of a line with one endpoint, going on endlessly in one direction
Opposite rays	Two rays from one point going opposite ways along the same line
Congruent angles	Angles with the same measure
Acute angle	Less than 90°
Right angle	Exactly 90°
Obtuse angle	More than 90° and less than 180°
Straight angle	Exactly 180°
Reflex angle	More than 180° and less than 360°

Count opposite rays point by point: at each crossing point, one pair for every line through it.

Common traps

Counting one pair per line instead of per point

A line through two marked crossing points gives a pair of opposite rays at each of them, so it counts twice. Go point by point and add.

Reading congruent as supplementary

Congruent angles are equal. An angle congruent to

120^\circ

is

120^\circ

, not

60^\circ

.

Mixing up complement and supplement

Complementary angles add to

90^\circ

, supplementary angles to

180^\circ

. The complement of

30^\circ

is

60^\circ

;

150^\circ

is its supplement.

Answering with the partner angle

If an angle is five times its supplement,

x = 5(180 - x)

gives

x = 150^\circ

and a supplement of

30^\circ

. The question asks for the angle, so the answer is

150^\circ

.

Intersecting Lines, Parallel Lines and Polygons

Learn this subtopic in the notes

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}

$n$ number of sides

Common traps

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

.

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.

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

.

Constructions

Learn this subtopic in the notes

Dividing a segment in a ratio

Dividing in the ratio m to n

\text{points on the ray} = m + n \qquad AP = \frac{m}{m+n} \times AB, \quad PB = \frac{n}{m+n} \times AB

Angles you can build with a ruler and compass

Angle	How it is built
60°	The equilateral arc construction
120°	Two 60° steps along the same arc
90°	Bisect the angle between the 60° and 120° marks
30°	Bisect 60°
45°	Bisect 90°
15°	Bisect 30°
22½°	Bisect 45°
135°	90° plus half of the next 90°
67½°	Bisect 135°
150°	120° plus half of the last 60°
32½°	Not by these steps It is half of 65°, and 65° cannot be built from 60°, 90° and their halves.

If an angle comes from 60° and 90° by adding, subtracting and halving, you can build it.

Common traps

Marking only the larger number of points

A ratio 5 : 9 needs

5 + 9 = 14

points, not 9. The ray must hold both parts of the ratio.

Marking one point too few

The steps are counted from A, so

m + n

steps need

m + n

marked points. 'm + n minus 1' is a printed wrong option.

Thinking a half degree makes an angle impossible

22\frac{1}{2}^\circ

is half of

45^\circ

and

67\frac{1}{2}^\circ

is half of

135^\circ

, so both can be built.

Thinking every half angle can be built

Halving only works on an angle you can already build.

32\frac{1}{2}^\circ

is half of

65^\circ

, and

65^\circ

does not come from

60^\circ

and

90^\circ

.

Lines, Angles and Polygons formulas

Lines, Rays and Angle Pairs

Complementary and supplementary angles

Lines, rays, segments and kinds of angle

Common traps

Intersecting Lines, Parallel Lines and Polygons

Angles where two lines cross

Parallel lines cut by a transversal

Angles of a triangle and a polygon

Common traps

Constructions

Dividing a segment in a ratio

Angles you can build with a ruler and compass

Common traps

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