- Two lines in a plane do exactly one of two things: they intersect at a single point, or they are parallel and never meet.
- When two lines cross, they make two pairs of equal vertically opposite angles, and any two adjacent angles form a linear pair that adds to $180°$.
- A transversal cutting two lines creates eight angles grouped as corresponding, alternate-interior, alternate-exterior and co-interior angles.
- If the two lines are parallel, then corresponding angles are equal, alternate angles are equal, and co-interior angles are supplementary ($180°$).
- These angle facts also work in reverse — they are the standard tests to prove two lines are parallel.
- You learn to draw a parallel to a given line through an outside point using a ruler and set-square (or compasses), and to spot parallel and intersecting lines all around you.
- Weightage: ~5 marks/year — usually a "find the missing angle" figure problem and one reasoning/construction question.
1. Lines that meet and lines that do not
Take a flat sheet of paper — a plane. Draw any two straight lines on it. There are only two possibilities. Either the lines cross each other at some point, in which case we call them intersecting lines, or they run alongside each other forever without ever meeting, in which case they are parallel lines.
Two intersecting lines meet at exactly one point — they cannot meet at two points, because two straight lines that share two points would actually be the same line. Parallel lines, on the other hand, stay the same distance apart everywhere; the gap between them never widens or shrinks.
Everyday examples are everywhere: the two rails of a railway track, opposite edges of a ruler, the lines on ruled paper, and the bars of a window grille are all parallel. The hands of a clock at most moments, two roads meeting at a junction, and the blades of a pair of scissors are intersecting.
2. Angles formed when two lines intersect
When two lines cross at a point, four angles are formed around that point. Two important relationships appear immediately.
Linear pair
Two adjacent angles whose non-common arms form a straight line are called a linear pair. Because a straight line is a "straight angle" of $180°$, a linear pair always adds up to $180°$. Such angles are said to be supplementary.
Vertically opposite angles
The two angles that are "across" from each other at the crossing point are called vertically opposite angles. They are always equal.
Let two lines cross, making angles $\angle 1,\ \angle 2,\ \angle 3,\ \angle 4$ in order around the point. Then $\angle 1$ and $\angle 2$ form a linear pair, so $\angle 1 + \angle 2 = 180°$. Also $\angle 2$ and $\angle 3$ form a linear pair, so $\angle 2 + \angle 3 = 180°$. Comparing the two, $\angle 1 + \angle 2 = \angle 2 + \angle 3$, hence $\angle 1 = \angle 3$. The same reasoning gives $\angle 2 = \angle 4$.
3. The transversal — a line that cuts two lines
A transversal is a line that crosses two (or more) other lines at distinct points. When a transversal cuts two lines, it makes eight angles in all — four at each crossing. These eight angles are given special names depending on where they sit.
Imagine two horizontal lines with a slanting transversal cutting both. Number the four angles at the top crossing $1, 2, 3, 4$ and the four at the bottom crossing $5, 6, 7, 8$, going clockwise from top-left.
| Type of pair | What it means | Examples |
|---|---|---|
| Corresponding angles | Same position at each crossing (e.g. both top-left). | $\angle 1$ & $\angle 5$, $\angle 2$ & $\angle 6$ |
| Alternate interior angles | Between the two lines, on opposite sides of the transversal. | $\angle 3$ & $\angle 5$, $\angle 4$ & $\angle 6$ |
| Alternate exterior angles | Outside the two lines, on opposite sides of the transversal. | $\angle 1$ & $\angle 7$, $\angle 2$ & $\angle 8$ |
| Co-interior (allied) angles | Between the two lines, on the same side of the transversal. | $\angle 3$ & $\angle 6$, $\angle 4$ & $\angle 5$ |
So far these names are just labels — they apply whether or not the two lines are parallel. The magic happens once the lines are parallel.
4. Angle properties when the lines are parallel
When the two lines cut by the transversal are parallel, the eight angles fall neatly into just two values: a set of equal "acute" angles and a set of equal "obtuse" angles (unless the transversal is perpendicular, when all eight are $90°$). The precise statements are:
• each pair of corresponding angles is equal;
• each pair of alternate interior angles is equal;
• each pair of alternate exterior angles is equal;
• each pair of co-interior angles is supplementary (adds to $180°$).
You only need to remember one of these well; the rest follow from linear pairs and vertically opposite angles. For instance, if corresponding angles are equal, then since one of them is vertically opposite to an alternate angle, the alternate angles must be equal too.
A transversal cuts two parallel lines, and one of the eight angles measures $70°$. Find every other angle.
The angle $70°$ has a linear-pair partner of $180° - 70° = 110°$. Its vertically opposite angle is again $70°$. So at that crossing the four angles are $70°, 110°, 70°, 110°$. Because the lines are parallel, corresponding angles repeat at the second crossing, giving exactly the same four values there: $70°, 110°, 70°, 110°$. So four angles are $70°$ and four are $110°$.
5. Using the properties to find missing angles
Most exam questions give a figure with one or two known angles and ask for an unknown angle marked $x$. The strategy is always: identify which named pair connects the known and unknown angle, then apply the matching rule.
Lines $\ell$ and $m$ are parallel; a transversal makes an angle of $55°$ with $\ell$ on the interior. The alternate interior angle on $m$ is $x$. Find $x$.
Alternate interior angles between parallel lines are equal, so $x = 55°$.
Lines $p \parallel q$ are cut by a transversal. One co-interior angle is $120°$ and the other is $y$. Find $y$.
Co-interior angles are supplementary: $y = 180° - 120° = 60°$.
In a figure, $AB \parallel CD$. A transversal meets $AB$ making $\angle = 65°$. A second transversal from the same point creates an angle $x$ that is corresponding to a $40°$ angle at $CD$. The required angle is the difference. Find it.
The corresponding angle to $40°$ is $40°$ (parallel lines). The remaining angle along the straight line $AB$ is $180° - 65° - 40° = 75°$. So $x = 75°$. The key is to break a multi-line figure into one rule at a time.
6. Tests for parallel lines (the converse)
Every property above works both ways. If, when a transversal cuts two lines, we find that a pair of corresponding angles is equal (or alternate angles equal, or co-interior angles supplementary), then the two lines must be parallel. These converse statements are how we actually prove that lines are parallel.
• a pair of corresponding angles is equal, OR
• a pair of alternate interior angles is equal, OR
• a pair of co-interior angles is supplementary.
A transversal cuts two lines so that a pair of co-interior angles measure $108°$ and $72°$. Are the lines parallel?
Their sum is $108° + 72° = 180°$. Co-interior angles are supplementary, so by the converse the lines are parallel.
Alternate interior angles measure $50°$ and $55°$. Since $50° \ne 55°$, the alternate angles are unequal, so the lines are not parallel.
7. Drawing a line parallel to a given line
A classic construction: given a line $\ell$ and a point $P$ not on it, draw the line through $P$ parallel to $\ell$. The idea uses the "equal corresponding angles" test.
1. Take any point $Q$ on $\ell$ and join $PQ$. This $PQ$ is a transversal.
2. At $Q$, the transversal $PQ$ makes some angle with $\ell$. Copy that same angle at $P$, on the same side, so that the new angle and the old one are corresponding angles.
3. Extend the new arm both ways. Because the corresponding angles are equal, this new line is parallel to $\ell$ and passes through $P$. The set-square method simply slides a set-square along a ruler to copy the angle automatically.
This also shows a deep fact: through a point outside a line, there is exactly one line parallel to the given line.
8. Common mistakes to avoid
- Calling co-interior angles "equal" — they are supplementary, not equal.
- Applying parallel-line rules when the lines are not parallel. The angle equalities hold only for parallel lines.
- Confusing alternate interior with alternate exterior — interior means between the two lines.
- Forgetting that vertically opposite angles and linear pairs work at any intersection, parallel or not.
- Thinking two lines can intersect at more than one point.
9. Quick revision checklist
- Two lines either intersect at one point or are parallel.
- At an intersection: vertically opposite angles equal; linear pair $= 180°$.
- Transversal → eight angles: corresponding, alternate interior, alternate exterior, co-interior.
- Parallel lines: corresponding equal, alternate equal, co-interior supplementary.
- Converses are the tests for parallelism.
- One unique parallel through an external point.
- intersecting lines
- perpendicular lines
- parallel lines
- transversal lines
- supplementary
- equal
- complementary
- right angles
- $90°$
- $180°$
- $270°$
- $360°$
- bisector
- perpendicular
- transversal
- diagonal
- $4$
- $6$
- $8$
- $12$
- equal
- supplementary
- complementary
- unequal
- equal
- complementary
- supplementary
- right angles
- $65°$
- $25°$
- $115°$
- $90°$
- perpendicular
- parallel
- intersecting at $48°$
- the same line
- $70°$
- $110°$
- $80°$
- $90°$
- $0$
- exactly $1$
- exactly $2$
- infinitely many
- corresponding angles
- alternate interior angles
- co-interior angles
- alternate exterior angles
- all $90°$
- $45°$ each
- $90°, 45°, 45°$
- cannot be found
- $15°$
- $25°$
- $30°$
- $75°$
- vertically opposite angles is equal
- linear-pair angles is $180°$
- alternate interior angles is equal
- angles is acute
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