Lines and angles

What this heading covers

Every geometry question in either paper is built from four things: a point, a line, an angle, and the rule connecting two angles that sit together. Nothing here needs derivation. It needs the names held exactly, because the paper tests the boundary between two names far more often than it tests a calculation.

This is also the cheapest part of the mathematics section to secure. A candidate who can name an angle and apply a linear pair will answer in fifteen seconds what otherwise takes two minutes of guessing.

Point, line, ray, segment, plane

• A point marks position and has no size. It is named with a capital letter.
• A line extends without end in both directions. It has no endpoints and no length.
• A line segment is the part of a line between two endpoints. It has a definite length.
• A ray has one endpoint and continues without end in one direction.
• A plane is a flat surface extending without end in every direction.

The distinction the paper actually asks about is line against line segment against ray, usually by showing a figure or by asking how many of each appear in it. Count endpoints: two endpoints is a segment, one is a ray, none is a line.

Collinear points lie on one line. Concurrent lines pass through one point.

How two lines can sit

• Parallel lines never meet, however far they are extended. The distance between them stays equal.
• Intersecting lines meet at exactly one point.
• Perpendicular lines intersect at 90°.

Two distinct lines in a plane either intersect at one point or are parallel. They cannot meet twice — a question offering "two lines meeting at two points" is testing that.

The six angles, and where each one ends

Angle	Measure
Acute	more than 0° and less than 90°
Right	exactly 90°
Obtuse	more than 90° and less than 180°
Straight	exactly 180°
Reflex	more than 180° and less than 360°
Complete	exactly 360°

Read the boundaries, not just the names. 90° is a right angle, not acute and not obtuse; 180° is straight, not reflex. Questions are written precisely on those edges — "an angle of 90° is ___" has one correct answer and three that are nearly right.

An angle is measured with a protractor, in degrees. The two arms meet at the vertex.

Complementary and supplementary

• Complementary angles add to 90°.
• Supplementary angles add to 180°.

These two are swapped more often than anything else in this section. A way to keep them apart that survives exam pressure: C comes before S in the alphabet, and 90 comes before 180.

The complement of 30° is 60°. The supplement of 30° is 150°. An angle can have a supplement and no complement — anything of 90° or more.

The four relationships a figure is built from

Adjacent angles share a vertex and one arm, and do not overlap.

A linear pair is two adjacent angles whose outer arms form a straight line. They are always supplementary — they add to 180°. This is the single most used fact in the geometry part of either paper.

Vertically opposite angles are formed when two lines cross. They sit opposite each other at the crossing and are always equal. Two lines crossing make two such pairs.

A transversal is a line cutting two or more lines. When it cuts two parallel lines:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.
• Co-interior angles (also called allied or same-side interior) are supplementary, not equal.

That last line is where marks are lost. Three of the four pairs are equal and one adds to 180°, and the question usually chooses the one that adds to 180°.

The equalities hold only if the two lines are parallel. A figure that does not say the lines are parallel, and does not mark them so, is not giving you them.

A question of the kind the paper sets

Two angles form a linear pair. One is 4 times the other. Find both.

A linear pair adds to 180°. Let the smaller be x, so the larger is 4x.

x + 4x = 180° 5x = 180° x = 36°

The angles are 36° and 144°. Check: they add to 180°, and 144 = 4 × 36.

Notice what the question did not say. It did not say "supplementary" — it said "linear pair", and you had to know that a linear pair is supplementary. That substitution is the whole question.

Mistakes that cost marks here

• Swapping complementary and supplementary. The commonest error in this section.
• Using the parallel-line equalities when the lines are not given as parallel. Corresponding angles are equal only for parallel lines; otherwise they are merely two angles.
• Treating co-interior angles as equal. They are supplementary. Three pairs are equal and this one is not, so it is the one asked about.
• Calling 90° acute or obtuse. It is neither; it is right. Same for 180°, which is straight rather than reflex.
• Confusing "line" with "line segment" when counting in a figure, which changes the answer.
• Assuming adjacent angles are supplementary. They are supplementary only when they form a linear pair. Adjacent alone means sharing a vertex and an arm.

In one line

Hold the six angle boundaries exactly, keep complementary at 90° and supplementary at 180°, and remember that a linear pair is always supplementary while the parallel-line equalities apply only when the lines are actually parallel.

For the classroom

Children meet angles as turn before they meet them as a figure. Asking a class to face the window and then make a quarter turn, a half turn and a full turn builds right, straight and complete angles physically before any protractor appears, and it makes a reflex angle — the one children find strangest — simply a turn that went most of the way round.

For parallel lines and a transversal, two rulers and a pencil laid across them lets a child slide the pencil and watch which angles stay equal and which change. That is the difference between a remembered list and a rule they can reconstruct in the exam hall.