Basic Angle Facts and Vertically Opposite Angles

This rule applies independently of the number of angles; it works whether there are two, three, or twenty angles meeting at the vertex (the common point where the lines meet). If a point includes a reflex angle (an angle greater than $180^\circ$ but less than $360^\circ$), it is often the "remainder" of the full turn. Sometimes diagrams will include a square symbol at the vertex; this denotes a right angle of $90^\circ$, and you must include this $90^\circ$ in your sum even though the number is not written.

Worked Example 1: Numerical Calculation

Four angles meet at a point. Three of the angles are $110^\circ$, $95^\circ$, and $80^\circ$. Calculate the size of the fourth angle, $x$.

Step 1: State the rule.

• Angles at a point sum to $360^\circ$.

Step 2: Sum the known angles.

• $110^\circ + 95^\circ + 80^\circ = 285^\circ$

Step 3: Subtract from $360^\circ$ to find the unknown.

• $360^\circ - 285^\circ = 75^\circ$

Step 4: Final Answer.

• $x = 75^\circ$

Worked Example 2: Algebraic Problem

Three angles at a point are $x$, $2x$, and $120^\circ$. Calculate the value of $x$.

Step 1: Form the equation.

• $x + 2x + 120 = 360$

Step 2: Simplify the equation.

• $3x + 120 = 360$

Step 3: Subtract $120$ from both sides.

• $3x = 240$

Step 4: Divide by $3$.

• $x = 80^\circ$

Properties of Angles on a Straight Line

A straight line might not intuitively look like an angle, but geometrically it represents exactly half of a full $360^\circ$ rotation. Because a half-turn is half of a full turn, adjacent angles on a straight line (angles that share a common vertex and a common side, lying flat on a straight line) always sum to $180^\circ$.

When two angles specifically add up to $180^\circ$, they are known as supplementary angles. To apply this rule, you must ensure the base line is perfectly straight.

Worked Example 3: Angles on a Straight Line

A straight line $PQ$ has two angles meeting at a point: $54^\circ$ and $y$. Calculate the value of $y$.

Step 1: State the sum rule.

• Angles on a straight line sum to $180^\circ$.

Step 2: Substitute the known angles into the rule.

• $y + 54^\circ = 180^\circ$

Step 3: Subtract to find the final answer.

• $y = 180^\circ - 54^\circ = 126^\circ$

Properties of Vertically Opposite Angles

Understanding how lines intersect allows you to instantly find missing angles without doing any addition or subtraction. When two straight lines cross (intersect) at a single vertex, they form an "X" shape. The pairs of angles directly across from each other are called vertically opposite angles.

The defining property of vertically opposite angles is that they are always equal to each other. Every intersection of two straight lines produces exactly two pairs of these equal angles. Notice that angles adjacent to each other at the intersection (sharing a line but not opposite) are not vertically opposite; instead, they lie on a straight line and are supplementary ($180^\circ$).

Worked Example 4: Vertically Opposite Angles

Two straight lines intersect. The vertically opposite angles are given algebraically as $(6x + 10)^\circ$ and $(10x - 70)^\circ$. Calculate the value of $x$.

Step 1: Apply the property.

• Vertically opposite angles are equal.

Step 2: Form the equation.

• $6x + 10 = 10x - 70$

Step 3: Rearrange to solve for $x$.

• Subtract $6x$: $10 = 4x - 70$
• Add $70$: $80 = 4x$
• Divide by $4$: $x = 20$

Step 4: Check your answer.

• $6(20) + 10 = 130^\circ$
• $10(20) - 70 = 130^\circ$