SPI Mathematics · Free notes

Lines, Angles and Polygons: SPI Mathematics

Lines, Angles and Polygons has 19 questions in the SPI bank, and three of them are from the 2022 paper. Thirteen are EASY, and most of those test one word or one rule: what a ray is, what supplementary means, how many points divide a segment in a ratio. The marks are lost by mixing up 90 and 180 degrees, and by counting too few pairs in a figure, so these pages give each rule with the check that catches the slip.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

SPI home All formulas on one page

Subtopic notes

Formula & revision sheet

5 formulas · 2 reference tables · 14 gotchas across all subtopics — the exam-eve cheat-sheet

Lines, Rays and Angle Pairs

Formulas (1)

Complementary and supplementary angles · Complement and supplement $\text{complement of } x = 90^\circ - x \qquad \text{supplement of } x = 180^\circ - x$

Reference tables (1)

Lines, rays, segments and kinds of angle10 rows

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.

Watch out for (4)

Counting one pair per line instead of per point
→ Lines, rays, segments and kinds of angle
Reading congruent as supplementary
→ Lines, rays, segments and kinds of angle
Mixing up complement and supplement
→ Complementary and supplementary angles
Answering with the partner angle
→ Complementary and supplementary angles

Intersecting Lines, Parallel Lines and Polygons

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

Constructions

Formulas (1)

Dividing a segment in a ratio · Dividing in the ratio m to n $\text{points on the ray} = m + n \qquad AP = \frac{m}{m+n} \times AB, \quad PB = \frac{n}{m+n} \times AB$

Reference tables (1)

Angles you can build with a ruler and compass11 rows

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.

Watch out for (4)

Marking only the larger number of points
→ Dividing a segment in a ratio
Marking one point too few
→ Dividing a segment in a ratio
Thinking a half degree makes an angle impossible
→ Angles you can build with a ruler and compass
Thinking every half angle can be built
→ Angles you can build with a ruler and compass

Question weightage by concept

7 concepts · 19 questions — where the marks actually sit, so you know what to drill first

Lines, Rays and Angle Pairs9 questions · 47%

Concept	Qs	Share
Lines, rays, segments and kinds of angle	5	26%
Complementary and supplementary angles	4	21%

Intersecting Lines, Parallel Lines and Polygons5 questions · 26%

Concept	Qs	Share
Angles where two lines cross	2	11%
Parallel lines cut by a transversal	2	11%
Angles of a triangle and a polygon	1	5%

Constructions5 questions · 26%

Concept	Qs	Share
Dividing a segment in a ratio	3	16%
Angles you can build with a ruler and compass	2	11%

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper