Angles at a Point, on a Line and Vertically Opposite

• The $360^\circ$ Rule: $x_1 + x_2 + ... + x_n = 360^\circ$.
• In exam questions, you will often need to sum all the known angles around a point and subtract this total from $360^\circ$ to find the missing value.

Vertically Opposite Angles

Look at a pair of open scissors — the angle between the metal blades perfectly matches the angle between the plastic handles. When two straight lines intersect at a single point, they create an "X" shape. The non-adjacent angles opposite each other are called vertically opposite angles, and they are always equal in size.

• This rule only applies to perfectly straight intersecting lines. If a line bends at the intersection, the rule does not work.
• To prove this rule geometrically: if angle $a$ and angle $b$ sit on a straight line ($a + b = 180^\circ$), and angle $c$ and angle $b$ sit on another straight line ($c + b = 180^\circ$), then $a$ must equal $c$.

Calculating Missing Angles in Complex Diagrams

AQA exam diagrams usually include the warning "Not drawn accurately". This means you cannot measure angles with a protractor; you must calculate them using geometric rules. A good strategy is to "walk" through the diagram, finding any possible angle first to unlock the next step.

Worked Example: Combining Geometric Rules

Two straight lines, $AC$ and $BD$, intersect at point $O$. A ray, $OE$, also meets at $O$, splitting angle $AOD$. You are given the following information:

• Angle $AOB = (2x + 10)^\circ$
• Angle $COD = 70^\circ$
• Angle $AOE = 60^\circ$

Calculate $x$, find the adjacent angle $BOC$, and use the angles at a point rule to calculate the missing angle $EOD$.

Step 1: Calculate $x$ using vertically opposite angles.

• Angles $AOB$ and $COD$ are vertically opposite.
• $2x + 10 = 70$ (Reason: Vertically opposite angles are equal)
• $2x = 60 \Rightarrow x = 30$
• The value of $x$ is $30$, and angle $AOB$ is $70^\circ$.

Step 2: Calculate angle $BOC$ using a straight line.

• Angles $AOB$ and $BOC$ sit on the straight line $AC$.
• $180^\circ - 70^\circ = 110^\circ$
• Angle $BOC = 110^\circ$ (Reason: Angles on a straight line add up to $180^\circ$)

Step 3: Calculate angle $EOD$ using angles at a point.

• The angles around point $O$ are $AOB$, $BOC$, $COD$, $EOD$, and $AOE$.
• We know $AOB = 70^\circ$, $BOC = 110^\circ$, $COD = 70^\circ$, and $AOE = 60^\circ$.
• $70^\circ + 110^\circ + 70^\circ + EOD + 60^\circ = 360^\circ$ (Reason: Angles at a point add up to $360^\circ$)
• $310^\circ + EOD = 360^\circ$
• $EOD = 360^\circ - 310^\circ = 50^\circ$