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Lines, Angles and Pairs of Lines
Geometry begins with a few simple ideas: points, lines and angles. The APTET syllabus lists point, line, line segment, ray, curves, polygons, angles, measuring lines, pairs of lines, and the elements, measurement and types of angles. These terms are the vocabulary of every later geometry question, so precise definitions matter.
Basic geometrical ideas
| Term | Meaning | Key fact |
|---|---|---|
| Point | an exact position; it has no length, breadth or thickness | named by a capital letter: point A |
| Line | a straight path extending endlessly in both directions | no end points; cannot be measured |
| Line segment | part of a line between two points | two end points; has a definite length |
| Ray | part of a line starting at one point and going endlessly in one direction | one end point (the initial point) |
| Collinear points | points lying on the same line | three or more points may or may not be collinear |
| Plane | a flat surface extending endlessly | a sheet of paper is a model of part of a plane |
Through one point infinitely many lines can be drawn; through two distinct points exactly one line can be drawn.
Curves: any drawing made without lifting the pencil is a curve (in mathematics, a straight line is also a curve). A simple curve does not cross itself; a closed curve ends where it starts; an open curve does not. A polygon is a simple closed figure made only of line segments (triangle, quadrilateral, pentagon…).
Measuring a line segment: use a ruler (or a divider and ruler). Place the zero mark at one end, or measure between two marks and subtract. Keep the eye vertically above the mark to avoid parallax error. Units: 10 mm = 1 cm.
Angles: elements, measurement and types
An angle is formed by two rays with a common end point. The common point is the vertex and the rays are the arms. ∠ABC has vertex B. Angles are measured in degrees with a protractor; a complete turn is 360°.
| Type | Measure | Example |
|---|---|---|
| Zero angle | 0° | both arms together |
| Acute | between 0° and 90° | 45° |
| Right | exactly 90° | corner of a page |
| Obtuse | between 90° and 180° | 120° |
| Straight | exactly 180° | a straight line |
| Reflex | between 180° and 360° | 210° |
| Complete | 360° | one full turn |
Clock angles: the hour hand covers 30° each hour (360° ÷ 12). At 3:00 the angle is 90°; at 4:00 it is 4 × 30° = 120°.
Pairs of angles and pairs of lines
| Pair | Rule |
|---|---|
| Complementary angles | sum is 90° (complement of 35° is 55°) |
| Supplementary angles | sum is 180° (supplement of 65° is 115°) |
| Adjacent angles | common vertex, common arm, no common interior |
| Linear pair | adjacent angles whose non-common arms form a straight line; sum is 180° |
| Vertically opposite angles | formed when two lines intersect; they are equal |
Worked example: two angles form a linear pair and one is 4 times the other. x + 4x = 180°, so 5x = 180° and x = 36°; the angles are 36° and 144°.
Pairs of lines
- Intersecting lines meet at one point.
- Parallel lines lie in the same plane and never meet (railway tracks); written l ∥ m.
- Perpendicular lines meet at a right angle; written l ⊥ m.
A transversal is a line that cuts two or more lines at different points. When it cuts two parallel lines:
| Angle pair | Relation |
|---|---|
| Corresponding angles | equal |
| Alternate interior angles | equal |
| Alternate exterior angles | equal |
| Interior angles on the same side (co-interior) | supplementary (sum 180°) |
So if one co-interior angle is 70°, the other is 180° − 70° = 110°.
Exam focus
- Line: no end points; ray: one; segment: two.
- Reflex angles lie between 180° and 360°.
- Complementary = 90°, supplementary = 180°; vertically opposite angles are equal.
- Parallel lines and a transversal: corresponding and alternate angles equal; co-interior angles add to 180°.
- Clock: 30° per hour mark.
Example
Mrs Hemalatha gives Class 4 straws and pins. Children make angles, compare them with the corner of a notebook (a right angle) and sort them into “smaller than a corner” (acute) and “bigger than a corner” (obtuse). Only after this do they measure with a protractor.
Key takeaways
- A point has position only; a line has no end points; a segment has two; a ray has one.
- Exactly one line passes through two distinct points.
- Angles are acute, right, obtuse, straight, reflex or complete.
- Complementary angles total 90°; supplementary angles and linear pairs total 180°.
- Vertically opposite angles are equal.
- With parallel lines, corresponding and alternate angles are equal, co-interior angles are supplementary.
Practice
- A line segment has:
- (A) no end points
- (B) one end point
- (C) two end points
- (D) three end points
- The complement of 35° is:
- (A) 55°
- (B) 145°
- (C) 65°
- (D) 325°
- An angle of 210° is:
- (A) an obtuse angle
- (B) a straight angle
- (C) an acute angle
- (D) a reflex angle
- How many lines can pass through two distinct points?
- (A) infinitely many
- (B) exactly one
- (C) two
- (D) none
- The angle between the hands of a clock at 4:00 is:
- (A) 100°
- (B) 120°
- (C) 140°
- (D) 90°
- Two parallel lines are cut by a transversal. One interior angle on the same side of the transversal is 70°. The other is:
- (A) 70°
- (B) 20°
- (C) 90°
- (D) 110°
- Two lines intersect and one of the angles formed is 50°. The vertically opposite angle is:
- (A) 50°
- (B) 130°
- (C) 40°
- (D) 310°
- Two angles form a linear pair and one is four times the other. The smaller angle is:
- (A) 45°
- (B) 30°
- (C) 36°
- (D) 144°
Answers and explanations (tap to show)
- (C) A line segment has a definite length between two end points.
- (A) 90° − 35° = 55°.
- (D) 210° lies between 180° and 360°.
- (B) Exactly one line passes through two distinct points.
- (B) 4 hour marks × 30° = 120°.
- (D) Co-interior angles are supplementary: 180° − 70° = 110°.
- (A) Vertically opposite angles are equal.
- (C) x + 4x = 180°, so x = 36°.