Lines & Angles • Topic 2 of 3

Interior & Exterior Angles

The angle sum of any triangle is 180°, and of any n-sided polygon is (n − 2) × 180°. The exterior angle of a polygon, taken one per vertex, always sums to 360° no matter how many sides — a fact CAT loves because it makes a regular polygon question a one-line division: each exterior angle of a regular n-gon is 360°/n, so each interior angle is 180° − 360°/n. The exterior angle theorem for a triangle is the workhorse: an exterior angle equals the sum of the two remote (non-adjacent) interior angles, which instantly gives the third angle without finding the others. Keep the two regular-polygon equations ready — number of sides from a given interior angle is n = 360 / (180 − interior), and the interior-to-exterior ratio (e.g. 5 : 1) pins n down immediately. Most CAT polygon problems are really just these two facts dressed up in a diagram.

✅ Solved examples

1. Find the sum of the interior angles of a regular octagon, and each interior angle.
Sum = (8 − 2) × 180° = 1080°. Each = 1080/8 = 135°.
2. Each exterior angle of a regular polygon is 24°. How many sides does it have?
Exterior angles sum to 360°: n = 360/24 = 15 sides.
3. In a triangle, an exterior angle is 110° and one remote interior angle is 45°. Find the other remote interior angle.
Exterior = sum of remote interiors: 110° = 45° + x ⇒ x = 65°.
4. The interior angle of a regular polygon is 156°. Find the number of sides.
Exterior = 180 − 156 = 24°; n = 360/24 = 15 sides.

✏️ Practice — try these, take hints as needed

1. Find the sum of the interior angles of a 12-sided polygon (dodecagon).
Use (n − 2) × 180°.
n = 12.
10 × 180°.
1800°
2. Each interior angle of a regular polygon is 144°. Find the number of sides.
Exterior = 180 − 144.
= 36°.
n = 360/36.
10
3. An exterior angle of a triangle is 125° with remote interior angles in ratio 2 : 3. Find the larger remote angle.
Remote interiors sum to the exterior.
2k + 3k = 125.
k = 25, larger = 3k.
75°
4. The ratio of an interior angle to its exterior angle in a regular polygon is 5 : 1. Find the number of sides.
Interior + exterior = 180°.
5k + k = 180 ⇒ exterior = 30°.
n = 360/30.
12
5. How many sides does a regular polygon have if each exterior angle is 18°?
Exterior angles sum to 360°.
n = 360/exterior.
360/18.
20

📝 Topic test — 8 questions

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