SSC CGL Maths · Lines, Angles and Polygons

Angles, Parallels and Polygon Sums

Angles on a straight line add to 180°, angles round a point to 360°, a polygon with n sides has interior angles adding to (n − 2) × 180°, its exterior angles always add to 360°, and it has n(n − 3)/2 diagonals.

All Lines, Angles and Polygons notes

Why this matters

Eight questions in the bank, seven EASY and one MODERATE, all from the 2025 shifts. Each is one fact used once: a vertically opposite or alternate angle, an angle sum, a missing angle, a count of diagonals. Learn the three formulas until you can write them without thinking and each of these takes well under the 36 seconds you have.

Concept 1 of 3: Angles at crossing lines and at parallel lines

A straight line is half a turn, so the angles along it share 180°. A full turn round a point is 360°. When a third line crosses two parallel lines, it meets both at the same slant, so the angles it makes at one line repeat at the other.

Definition

Linear pair: two angles side by side on a straight line add to $180^\circ$ .
Angles round a point add to $360^\circ$ .
Vertically opposite angles (across the crossing point of two lines) are equal.
A transversal is a line that cuts two other lines. When those two lines are parallel: corresponding angles are equal, alternate interior angles are equal, and co-interior angles (between the parallels, on the same side of the transversal) add to $180^\circ$ .
The equal-angle rules work only for parallel lines. Read the reverse too: if alternate angles are equal, the lines are parallel.

Angles at lines

\text{linear pair} = 180^\circ \qquad \text{round a point} = 360^\circ \qquad \text{co-interior pair} = 180^\circ

Worked example

Two straight lines cross each other. One of the four angles formed is

40^\circ

more than the angle next to it. Find all four angles.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 20 Sep 2025, Shift 1 · Q69Easy

Example 1 · Lines, Angles and Polygons · Angles, Parallels and Polygon Sums

When a transversal intersects two parallel lines, one of the alternate interior angles is

115^\circ

. What is the measure of the angle that lies opposite to it on the alternate interior side?

Calling alternate angles supplementary

With parallel lines, alternate interior angles are equal. The pair that adds to

180^\circ

is the co-interior pair, on the same side of the transversal. If one alternate angle is

115^\circ

, the other is

115^\circ

, and

65^\circ

is a wrong option.

Using the parallel-line rules on lines that are not parallel

Corresponding and alternate angles are equal only when the two lines cut by the transversal are parallel. If the question does not say so, or give equal angles that prove it, do not assume it.

Drill 1 more on angles at crossing lines and at parallel lines

Concept 2 of 3: Interior and exterior angle sums of a polygon

Pick one corner of a polygon and join it to every other corner. An n-sided polygon splits into n − 2 triangles, and each triangle holds 180°. For the outside angles, walk once round the edge: at each corner you turn through the exterior angle, and one lap is one full turn, 360°, however many corners there are.

Definition

Interior angle sum of an n-sided polygon $= (n - 2) \times 180^\circ$ .
Exterior angle sum, one at each corner, $= 360^\circ$ for every convex polygon. It does not depend on n.
At each corner, interior angle $+$ exterior angle $= 180^\circ$ (a linear pair).
Sums worth knowing by heart: triangle $180^\circ$ , quadrilateral $360^\circ$ , pentagon $540^\circ$ , hexagon $720^\circ$ , heptagon $900^\circ$ , octagon $1080^\circ$ . Each extra side adds $180^\circ$ .
None of this needs the polygon to be regular.

Polygon angle sums

\text{interior sum} = (n - 2) \times 180^\circ \qquad \text{exterior sum} = 360^\circ \qquad \text{interior} + \text{exterior} = 180^\circ

nthe number of sides, which is also the number of corners

Worked example

The five interior angles of a pentagon are

x

x + 10

x + 20

x + 30

and

x + 40

degrees. Find the largest angle.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 3 · Q70Moderate

Example 2 · Lines, Angles and Polygons · Angles, Parallels and Polygon Sums

The sum of the measures of

7

interior angles of an

8

-sided polygon is

1,000^\circ

. What is the measure of the

8

th exterior angle?

Using n × 180° for the interior sum

The interior sum is

(n - 2) \times 180^\circ

, because the polygon splits into

n - 2

triangles, not n. A decagon's angles add to

1440^\circ

, not

1800^\circ

Thinking the exterior sum grows with the sides

One exterior angle at each corner of any convex polygon adds to

360^\circ

, for a triangle and for a 20-sided figure alike. 'Depends on the number of sides' is a printed wrong option.

Giving the interior angle when the exterior one is asked

When a question finds a missing interior angle and then asks for the exterior angle at that corner, take one more step: exterior

= 180^\circ -

interior. The interior value is usually printed as a wrong option.

Drill 3 more on interior and exterior angle sums of a polygon

Concept 3 of 3: Number of diagonals of a polygon

A diagonal joins two corners that are not next to each other. From one corner you can reach every other corner except itself and its two neighbours, so n − 3 diagonals start there. Doing that at all n corners counts every diagonal twice, once from each end.

Definition

Diagonals from one corner $= n - 3$ .
Total diagonals $= \frac{n(n - 3)}{2}$ .
Another way: every pair of corners is joined by a side or a diagonal. There are $\frac{n(n - 1)}{2}$ pairs, and n of them are sides, so the diagonals are $\frac{n(n - 1)}{2} - n$ .
Fast route when the diagonals are given and n is asked: try the options in $n(n - 3) = 2 \times$ diagonals.

Diagonals of an n-sided polygon

\text{diagonals} = \frac{n(n - 3)}{2}

Worked example

How many diagonals does a polygon with 10 sides have?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 2 · Q57Easy

Example 3 · Lines, Angles and Polygons · Angles, Parallels and Polygon Sums

The sum of the interior angles of a regular polygon is six times the sum of its exterior angles. Determine the number of diagonals in the polygon.

Counting every pair of corners as a diagonal

\frac{n(n - 1)}{2}

counts every pair of corners, sides included. Take away the n sides: a hexagon has

15 - 6 = 9

diagonals, not 15.

Forgetting to halve

n(n - 3)

counts each diagonal from both ends. Divide by 2: a hexagon has

\frac{6 \times 3}{2} = 9

diagonals, not 18.

Drill 1 more on number of diagonals of a polygon

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Angles at crossing lines and at parallel lines
Angles at lines
$\text{linear pair} = 180^\circ \qquad \text{round a point} = 360^\circ \qquad \text{co-interior pair} = 180^\circ$
Interior and exterior angle sums of a polygon
Polygon angle sums
$\text{interior sum} = (n - 2) \times 180^\circ \qquad \text{exterior sum} = 360^\circ \qquad \text{interior} + \text{exterior} = 180^\circ$
Number of diagonals of a polygon
Diagonals of an n-sided polygon
$\text{diagonals} = \frac{n(n - 3)}{2}$

Watch out for (7)

Calling alternate angles supplementary
→ Angles at crossing lines and at parallel lines
Using the parallel-line rules on lines that are not parallel
→ Angles at crossing lines and at parallel lines
Using n × 180° for the interior sum
→ Interior and exterior angle sums of a polygon
Thinking the exterior sum grows with the sides
→ Interior and exterior angle sums of a polygon
Giving the interior angle when the exterior one is asked
→ Interior and exterior angle sums of a polygon
Counting every pair of corners as a diagonal
→ Number of diagonals of a polygon
Forgetting to halve
→ Number of diagonals of a polygon

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