Lines, Angles and Parallel Lines
Lines, Angles and Parallel Lines
Geometry and mensuration together are the largest block in the Part-III maths paper, worth 10 to 12 of the 40 questions. This first topic is pure definition and angle relationship, and it is the base every later proof rests on, so nothing here should be left approximate.
Types of Angles
| Angle | Measure |
|---|---|
| Acute | Less than 90° |
| Right | Exactly 90° |
| Obtuse | Between 90° and 180° |
| Straight | Exactly 180° |
| Reflex | Between 180° and 360° |
| Complete | Exactly 360° |
- Complementary angles add to 90 degrees.
- Supplementary angles add to 180 degrees.
- Adjacent angles share a vertex and an arm but no interior points.
- A linear pair is two adjacent angles whose non-common arms form a straight line; they are supplementary.
- Vertically opposite angles are formed by two intersecting lines and are always equal.
✗ Complementary angles add to 180° | ✓ Complementary add to 90°; supplementary add to 180°
The C-before-S order helps: C for complementary comes before S for supplementary, and 90 comes before 180.
A Transversal Cutting Parallel Lines
When a transversal cuts two parallel lines, eight angles are formed and only two distinct values appear among them.
| Pair | Relationship |
|---|---|
| Corresponding angles | Equal |
| Alternate interior angles | Equal |
| Alternate exterior angles | Equal |
| Co-interior (allied) angles | Supplementary, add to 180° |
| Vertically opposite angles | Equal |
Only the co-interior pair is supplementary; every other named pair is equal. That is the whole of the topic, and the whole of the trap, since options routinely offer supplementary where equal is correct.
The converse also holds and is asked: if a pair of alternate interior angles is equal, the lines must be parallel. This is the standard way a question asks you to prove parallelism.
Angle Sums to Memorise
Triangle: 180° | Quadrilateral: 360° | Polygon of n sides: (n − 2) × 180°
Exterior angle sum of any polygon = 360°, whatever the number of sides
The exterior angle sum being constant surprises students and is therefore asked often. A regular polygon of n sides has each exterior angle equal to 360/n, which is the fastest way to find the number of sides when an angle is given.
If each exterior angle is 24 degrees, then n = 360/24 = 15 sides. Working through the interior angle instead takes three times as long.
समांतर रेखाओं में केवल सह-अंतर्कोण ही संपूरक होते हैं, बाकी सब बराबर।
TRE pointer: Geometry carries the heaviest weight in Part-III mathematics, roughly 10 to 12 questions against barely one in Part-II, so this is where the subject paper is won. The exterior angle sum of 360 degrees for every polygon is the single most useful fact in the block. Expect at least one question giving an exterior angle and asking for the number of sides. With five options and −1/3 negative marking, always draw the figure before choosing.
60-Second Recap
- Complementary add to 90°, supplementary to 180°; a linear pair is supplementary.
- Vertically opposite angles are always equal.
- With parallel lines, only co-interior angles are supplementary; all other named pairs are equal.
- Equal alternate interior angles prove the lines are parallel.
- Angle sum of an n-sided polygon is (n − 2) × 180°.
- Exterior angle sum is 360° for every polygon; each exterior angle of a regular one is 360/n.
Frequently Asked Questions
Which angle pair is supplementary when a transversal cuts parallel lines?
Only the co-interior, also called allied, pair. Corresponding angles, alternate interior angles, alternate exterior angles and vertically opposite angles are all equal. Options in these questions deliberately offer supplementary in place of equal, so the distinction is worth fixing firmly.
What is the sum of exterior angles of a polygon?
Three hundred and sixty degrees, for every polygon regardless of the number of sides. This surprises most students, since the interior angle sum grows with the number of sides. For a regular polygon each exterior angle is therefore three hundred and sixty divided by n.
How do I find the number of sides from an exterior angle?
Divide three hundred and sixty by the given exterior angle. An exterior angle of twenty-four degrees means fifteen sides. Going through the interior angle formula gives the same answer but takes far longer, which matters across a forty-question paper.
How do I prove two lines are parallel?
Show that one pair of corresponding angles is equal, or one pair of alternate interior angles is equal, or that a co-interior pair adds to a hundred and eighty degrees. Any one of these converse statements is enough, and questions usually supply exactly one of them.