SPI Mathematics · Lines, Angles and Polygons

Lines, Rays and Angle Pairs

A line has no endpoints, a ray has one and a segment has two; two angles are complementary when they add to 90 degrees and supplementary when they add to 180 degrees.

All Lines, Angles and Polygons notes

Why this matters

Nine questions in the bank, seven of them EASY, and all three of this chapter's questions from the 2022 paper are here. Most are one definition: what a line is, what congruent means, the complement of 30 degrees. The two harder ones from the 2022 paper are counting questions, and they are lost by counting too few.

Concept 1 of 2: Lines, rays, segments and kinds of angle

Geometry questions use a small set of words very exactly. A line, a ray and a segment differ only in their endpoints, and an angle is named by how it compares with 90 and 180 degrees.

Definition

Two facts settle the counting questions:

Opposite rays start at the same point and go in opposite directions along one line. Every line through a point gives one pair of opposite rays at that point.
Congruent angles have equal measures, so an angle congruent to $55^\circ$ is $55^\circ$ .

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.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q8Moderate

Example 1 · Lines, Angles and Polygons · Lines, Rays and Angle Pairs

Line AB, line CD, line LM, line ST and line EF intersect each other at distinct points as shown in the figure. Find the number of pairs of opposite rays formed.

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

120^\circ

, not

60^\circ

Drill 4 more on lines, rays, segments and kinds of angle

Concept 2 of 2: Complementary and supplementary angles

Two angles that fill a right angle together are complementary; two that fill a straight angle are supplementary. Name the unknown angle x, write its partner as 90 minus x or 180 minus x, and the word problem becomes one equation.

Definition

Complementary: the two angles add to $90^\circ$ . The complement of $x$ is $90^\circ - x$ .
Supplementary: the two angles add to $180^\circ$ . The supplement of $x$ is $180^\circ - x$ .
Supplementary angles need not sit side by side. Two neighbouring angles on a straight line are a linear pair, but any two angles adding to $180^\circ$ count, so a figure can have more supplementary pairs than linear pairs.
When the value you find is not among the options, the answer is 'None of these'.

Complement and supplement

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

Worked example

An angle is twice its complement. Find the angle.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q6Easy

Example 2 · Lines, Angles and Polygons · Lines, Rays and Angle Pairs

If measure of an angle is four times the measure of its supplementary angle, then measure of the angle is____deg.

Mixing up complement and supplement

Complementary angles add to

90^\circ

, supplementary angles to

180^\circ

. The complement of

30^\circ

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

Drill 3 more on complementary and supplementary angles

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 (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

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 Drill all 9 questions