Basic Angle Facts and Vertically Opposite Angles

Angles around a point

An angle measures a turn. One complete turn brings you back to the direction you started from and measures $360°$. So all the angles meeting at a point add up to $360°$, however many angles there are. If three of them are $75°$, $110°$ and $x$, then $75°+110°+x=360°$, giving $x=175°$.

Angles on a straight line

A straight line makes half a turn, or $180°$. When a line or ray divides that straight angle into smaller angles, those angles add up to $180°$. In the Edexcel sample diagram, $X$, $Y$ and $V$ lie on a straight line. The angle inside the quadrilateral at $Y$ and the adjacent angle therefore total , so the inside angle is . The reason is the straight-line angle fact, not the appearance of the drawing.

A right angle is $90°$: it is half a straight angle. Lines that meet at a right angle are perpendicular. A small square at their intersection marks the right angle, and each of the four angles made by two perpendicular straight lines is $90°$.

Vertically opposite angles

When two straight lines cross, they make four angles. The angles directly across the intersection from one another are vertically opposite, and each pair is equal. To see why, suppose an angle is $62°$. Either angle next to it makes a straight line with it, so each is $180°-62°=118°$. The angle opposite the original $62°$ is next to a $118°$ angle, so it must also be $$.

In a multi-step problem, first decide which lines are genuinely straight and which angles share a vertex. You can then use $180°$ for a straight line, $360°$ for a full turn and equality for vertically opposite angles, recording a reason for each deduction.