A complete guide to angle facts, parallel-line angle rules, triangle and quadrilateral properties, and the interior/exterior angle formulas for any polygon.
Geometry is the mathematics of shape, space and position. Almost everything you need — from the angles in a triangle to the exterior angle of a twelve-sided polygon — follows from a small handful of basic angle facts, applied carefully and combined step by step. This guide builds geometry from those first principles: angles on a line and at a point, through triangles and quadrilaterals, up to the general interior and exterior angle formulas for any polygon, and the symmetry properties that tie shapes together.
Three facts underpin almost all angle work. Angles on a straight line add up to 180°. Angles around a point add up to 360°. When two straight lines cross, the angles opposite each other are called vertically opposite angles, and they are always equal.
Three angles meet at a point on a straight line: 55°, 75° and x°. Since they lie on a straight line, 55 + 75 + x = 180, so x = 50°.
Four angles meet at a single point (not on a line): 90°, 130°, 70° and y°. Since angles at a point sum to 360°, y = 360 − 290 = 70°.
When a straight line (a transversal) crosses two parallel lines, it creates three important angle pairs. Corresponding angles (in matching positions at each intersection) are equal. Alternate angles (on opposite sides of the transversal, between the parallel lines, forming a "Z" shape) are equal. Co-interior angles (also called allied angles, on the same side of the transversal, between the parallel lines, forming a "C" shape) sum to 180°.
Two parallel lines are cut by a transversal. One angle at the first intersection is 108°. The corresponding angle at the second intersection is also 108°. The alternate angle (on the opposite side, between the lines) is also 108°. The co-interior angle on the same side, between the lines, is 180 − 108 = 72°.
The angles inside any triangle always add up to 180°. This single fact, combined with the basic angle rules above, unlocks most triangle problems. Triangles are classified by their sides (scalene: no equal sides; isosceles: two equal sides; equilateral: three equal sides) and by their angles (acute: all angles under 90°; right-angled: one angle exactly 90°; obtuse: one angle over 90°).
A triangle has angles of 62° and 79°. The third angle is 180 − 62 − 79 = 39°, so the triangle is acute-angled and scalene (all three angles, and therefore all three sides, are different).
If one side of a triangle is extended, the angle formed outside the triangle is its exterior angle. A key theorem states that the exterior angle of a triangle equals the sum of the two interior opposite angles (the two angles not adjacent to it). This is a shortcut that avoids needing the third interior angle directly.
A triangle has interior angles of 48° and 65° at two vertices. The exterior angle at the third vertex is 48 + 65 = 113°. As a check, the interior angle at that vertex is 180 − 113 = 67°, and 48 + 65 + 67 = 180 ✓.
In an isosceles triangle, the two sides that are equal sit opposite two equal base angles. In an equilateral triangle, all three sides are equal and all three angles are equal, so each angle is 180 ÷ 3 = 60°. These properties are used constantly in angle-chasing problems, especially when a diagram marks two sides with identical tick marks.
When you see two sides marked equal on a diagram, immediately mark the two angles opposite them as equal too — this single step often unlocks the rest of an angle-chasing question.
Any quadrilateral (four-sided shape) can be split into two triangles by a single diagonal, so its interior angles always sum to 2 × 180° = 360°. Common special quadrilaterals include the square, rectangle, parallelogram, rhombus, trapezium and kite.
A quadrilateral has angles of 95°, 80° and 110°. The fourth angle is 360 − 95 − 80 − 110 = 75°.
| Quadrilateral | Sides | Angles | Diagonals |
|---|---|---|---|
| Parallelogram | Opposite sides equal & parallel | Opposite angles equal; co-interior angles sum to 180° | Bisect each other |
| Rectangle | Opposite sides equal & parallel | All angles 90° | Equal, bisect each other |
| Rhombus | All four sides equal | Opposite angles equal | Bisect each other at 90° |
| Square | All four sides equal | All angles 90° | Equal, bisect each other at 90° |
| Trapezium | One pair of sides parallel | Co-interior angles on the parallel sides sum to 180° | Not generally equal |
| Kite | Two pairs of adjacent equal sides | One pair of opposite angles equal | Cross at 90°; one is bisected |
A polygon with n sides can be split into (n − 2) triangles by drawing diagonals from one vertex. Since each triangle contributes 180°, the sum of the interior angles of any polygon is:
A hexagon has n = 6 sides. Interior angle sum = (6 − 2) × 180 = 4 × 180 = 720°.
If the hexagon is regular (all sides and angles equal), each interior angle = 720 ÷ 6 = 120°.
The exterior angles of any convex polygon, taken one at each vertex, always sum to 360° — this is true whatever the number of sides. For a regular polygon with n sides, each exterior angle = 360 ÷ n, and each interior angle = 180° − exterior angle (since interior and exterior angles at a vertex lie on a straight line).
A regular polygon has an exterior angle of 24°. The number of sides is n = 360 ÷ 24 = 15 sides. Each interior angle is 180 − 24 = 156°.
A shape has line symmetry if a mirror line can be drawn so that one half is the exact reflection of the other. A shape has rotational symmetry of order n if it looks identical n times during a full 360° turn (n ≥ 2). A regular polygon with n sides always has n lines of symmetry and rotational symmetry of order n.
A regular pentagon (5 equal sides) has 5 lines of symmetry and rotational symmetry of order 5 (it maps onto itself every 360 ÷ 5 = 72° of rotation).
Most exam-style geometry questions combine several rules in one diagram. The reliable strategy is to work through a diagram systematically: mark every angle you can find immediately from a single rule, then use those new angles to unlock further ones, repeating until the required angle is found.
Triangle ABC is isosceles with AB = AC, and angle BAC = 42°. Line BC is extended to point D. Find the exterior angle ACD.
Since AB = AC, angles ABC and ACB are equal. They sum to 180 − 42 = 138°, so each is 138 ÷ 2 = 69°.
By the exterior angle theorem, angle ACD = angle BAC + angle ABC = 42 + 69 = 111°.
Common errors include: assuming lines are parallel or equal without a marking or statement confirming it; confusing alternate angles ("Z" shape) with co-interior angles ("C" shape); forgetting that the exterior angle sum of 360° applies to any convex polygon, not just triangles; and using the interior angle sum formula with the wrong value of n (n is the number of sides, not the number of diagonals).
| Rule | Statement |
|---|---|
| Angles on a line | Sum to 180° |
| Angles at a point | Sum to 360° |
| Vertically opposite angles | Equal |
| Alternate angles | Equal ("Z" shape) |
| Co-interior angles | Sum to 180° ("C" shape) |
| Triangle angle sum | 180° |
| Exterior angle of a triangle | Equals the sum of the two interior opposite angles |
| Quadrilateral angle sum | 360° |
| Polygon interior angle sum | (n − 2) × 180° |
| Polygon exterior angle sum | 360° (always) |
Nearly every geometry problem you meet at this level is a combination of these ten facts. Learn them well, and the rest becomes a matter of careful, systematic reasoning through a diagram.
This article is original EDUSAMBAM educational writing. It is designed as a broad geometry resource covering the core angle, triangle, quadrilateral and polygon properties needed for confident geometric reasoning. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: learn the ten core rules in the summary table, practise angle-chasing diagrams step by step, then attempt multi-step problems that combine several rules at once.
20 questions covering angle facts, parallel lines, triangles, quadrilaterals and polygons. Answer every question, then submit to see your score instantly.