Cracking the Angle Code: Rules for 2D Shapes

Imagine folding a piece of paper perfectly flat. The crease makes a straight line. Now draw two rays shooting out from the same point on that line — they slice the straight line into two angles. Here is the surprising thing: no matter how you tilt those rays, the two angles always add up to exactly the same total. What do you think that total is? Hold your answer in your head — we are about to prove it.

Angles are measured in degrees (°). A full turn is 360°, a half turn — a straight line — is 180°. These facts lead directly to three powerful rules. **Rule 1 — Angles on a straight line sum to 180°.** When two or more angles sit next to each other on one side of a straight line and share the same vertex (corner point), they always add up to 180°. This is sometimes called the 'supplementary' rule. For example, if one angle is 110°, the other must be 180° − 110° = 70°. **Rule 2 — Angles in a triangle sum to 180°.** Every triangle, no matter its shape or size, has three interior angles (angles inside the shape). Add them together and the total is always 180°. This is one of the most useful facts in geometry. If two angles in a triangle are 90° and 35°, the third must be 180° − 90° − 35° = 55°. **Rule 3 — Angles in a quadrilateral sum to 360°.** A quadrilateral is any four-sided polygon (closed flat shape with straight sides). Its four interior angles always add up to 360°. You can see why: a quadrilateral can always be cut into two triangles, each contributing 180°, giving 2 × 180° = 360°. If three angles in a quadrilateral are 85°, 100°, and 90°, the fourth must be 360° − 85° − 100° − 90° = 85°. When a question uses the notation '∠ABC', it means 'the angle at vertex B, between rays BA and BC'. The middle letter is always the vertex.