Points, lines and angles

Plane geometry is the science of figures in a plane. It rests on undefined basic notions (point, line, plane) and axioms; every other notion is defined through them and every theorem is proved from them. The properties of segments, angles and parallel lines in this topic are the “language” of all later geometry problems: in problems on triangles, quadrilaterals and circles they are used at every step. So it is important not only to memorise each rule but to understand why it is true.

§1. Basic figures

• A line is infinite; through two points there passes exactly one line. Two different lines have at most one common point.
• A ray is the part of a line going from one point (its start) infinitely in one direction.
• A segment is the part of a line between two points; it has a length.
• The midpoint of a segment halves it. If C lies on the segment AB, then AC + CB = AB.
• Through n points, no three of which are collinear, one can draw n(n − 1)/2 lines: each pair of points gives one line.

If a problem does not say how the points are arranged, consider every case: if A, B, C are collinear, B may lie between A and C or at the end.

Worked example

AB = 12, point C divides AB in the ratio AC : CB = 1 : 3, and M is the midpoint of AB. Find CM.

1. AC = 12 · 1/4 = 3.
2. AM = 12 : 2 = 6.
3. CM = AM − AC = 6 − 3 = 3.

Worked example

Points A, B, C are collinear, AB = 7, BC = 3. Find AC.

1. If C is on the other side of B from A: AC = 7 + 3 = 10.
2. If C lies between A and B: AC = 7 − 3 = 4.
3. Answer: 10 or 4.

§2. Angles

An angle is formed by two rays from one point (the vertex). Angles are measured in degrees; a straight angle is 180°. Types: acute (< 90°), right (90°), obtuse (between 90° and 180°), straight (180°).

Adjacent angles on a line share one arm while the other two arms form a line, so together they make a straight angle. Vertically opposite angles are equal because both are adjacent to the same angle: from α + β = 180° and γ + β = 180° we get α = γ.
An angle bisector halves the angle. The bisectors of two adjacent angles on a line are perpendicular: α/2 + β/2 = 180°/2 = 90°.
n rays from one point form n(n − 1)/2 angles (counting those smaller than a straight angle).

Worked example (official sample)

Two intersecting lines form adjacent angles whose difference is 20°. Find the smaller one.

1. Let the smaller be x and the larger x + 20°; they are adjacent: x + (x + 20°) = 180°.
2. 2x = 160° → x = 80°.

Worked example

Find the angle between the hands of a clock at 3:40.

1. The minute hand moves 6° per minute: 40 · 6° = 240°.
2. The hour hand moves 30° per hour and 0.5° per minute: 3 · 30° + 40 · 0.5° = 110°.
3. Difference: 240° − 110° = 130°.

§3. Parallel lines

Two lines in a plane that never meet are parallel. The parallel axiom: through a point off a line there is only one line parallel to it.
When a third line (a transversal) cuts two parallel lines, 8 angles are formed:

Conversely, if any one of these holds, the lines are parallel — these are the tests for parallel lines. Two lines perpendicular to a third are parallel.
In practice the 8 angles take only two values — α and 180° − α. Knowing this, any angle can be found at a glance.

Worked example

With two parallel lines and a transversal, the co-interior angles are in the ratio 2 : 7. Find them.

1. The angles are 2x and 7x, adding to 180°.
2. 9x = 180° → x = 20°.
3. The angles are 40° and 140°.

§4. The angle sum of a triangle

One of the most important facts of geometry follows from parallel lines. Draw through a vertex a line parallel to the opposite side: since alternate angles are equal, the three angles of the triangle together form a straight angle.

Consequences: a triangle has at most one right or obtuse angle; each angle of an equilateral triangle is 60°.

Worked example

An exterior angle of a triangle is 120°, and the two interior angles not adjacent to it are in the ratio 1 : 3. Find them.

1. x + 3x = 120° (exterior angle property).
2. x = 30°, the other is 90°.
3. The third angle: 180° − 120° = 60°.

§5. Perpendiculars and distance

The perpendicular from a point to a line is the shortest distance; any oblique line from that point is longer (in a right-angled triangle the hypotenuse is longer than a leg).

• Every point of the perpendicular bisector of a segment is equidistant from its ends (and conversely).
• Every point of an angle bisector is equidistant from the arms of the angle.

These two properties are used later to find the centres of a triangle’s circumcircle and incircle.

Worked example

From point A a perpendicular AH = 5 and an oblique AB of length 13 are drawn to line a. Find HB and state the distance from A to the line.

1. AHB is a right-angled triangle with hypotenuse AB.
2. HB = √(13² − 5²) = √144 = 12.
3. The distance from A to a is AH = 5, not the oblique length 13.

Check yourself

1. One of two adjacent angles on a line is 4 times the other. Find the larger.

2. Two vertically opposite angles add up to 100°. What is each?

3. How many lines pass through 6 points, no three collinear?

4. With parallel lines an alternate angle is 65°. The co-interior angle?

5. The angle between the clock hands at 2:00.

6. The angle between the bisectors of two adjacent angles on a line.

7. Two angles of a triangle are 47° and 68°. The third?

8. How many angles do 5 rays from one point form?

Common mistakes

• Confusing adjacent and vertically opposite angles.
• Ignoring the hour hand’s movement during the minutes (0.5° per minute) in clock problems.
• Taking co-interior angles as equal — they add up to 180°.
• Forgetting that a point may lie on the extension of a segment rather than on it.
• Adding an exterior angle to its adjacent interior angle and getting more than 180°.
• Taking an oblique line as the distance from a point to a line — distance is measured along the perpendicular.