Angles formed at a point

When lines and rays meet at a single point, they carve up the plane around that point into angles. Some of those angles have special names and special relationships.

Idea

Adjacent angles share a common vertex and a common arm, and lie on opposite sides of that arm. Imagine slicing a piece of cake from the centre , two neighbouring slices share an edge.

Linear pair. When two adjacent angles together form a straight line, they make a linear pair. Their measures add to $180^\circ$ (a straight angle). So if one is $50^\circ$, the other must be $130^\circ$.

Vertically opposite angles. When two lines cross, four angles form. Opposite angles (like a bow-tie) are called vertically opposite. A beautiful fact: vertically opposite angles are always equal. (Proof: each is supplementary to the same adjacent angle.)

So at any intersection of two lines, there are really only two distinct angle values , call them $a$ and $b$ , with $a + b = 180^\circ$. The four angles come as $a, b, a, b$ going round.

Angles around a point. All the angles around a single point add up to $360^\circ$ (one full turn). So if three rays meet at a point and form angles of $120^\circ, 110^\circ$ and an unknown $x$, then $x = 360 - 120 - 110 = 130^\circ$.

Worked examples

Example 1. Two adjacent angles on a straight line are $\angle a$ and $\angle b$ with $\angle a = 65^\circ$. Find $\angle b$.

Linear pair: $a + b = 180^\circ$. So $b = 180 - 65 = 115^\circ$.

Example 2. Two lines intersect; one angle is $40^\circ$. Find the other three.

Vertically opposite to $40^\circ$ is another $40^\circ$. The other pair forms a linear pair with $40^\circ$, so each is $180 - 40 = 140^\circ$. Going round: $40^\circ, 140^\circ, 40^\circ, 140^\circ$.

Example 3. Three rays at a point form angles of $90^\circ$, $130^\circ$ and $x$. Find $x$.

Around a point sum is $360^\circ$. So $x = 360 - 90 - 130 = 140^\circ$.

Example 4. Two lines cross. One angle is twice the next adjacent angle. Find both.

Let one be $x$ and adjacent be $2x$. Linear pair: $x + 2x = 180^\circ \Rightarrow 3x = 180 \Rightarrow x = 60^\circ$. So the angles are $60^\circ$ and $120^\circ$ (alternating).

Try it yourself

1. Two adjacent angles on a straight line are $x$ and $3x$. Find both.
2. At an intersection one angle measures $75^\circ$. Find the others.
3. Three rays at a point form angles of $80^\circ$, $100^\circ$, $x$. Find $x$.
4. Are two angles of $70^\circ$ and $110^\circ$ a linear pair? Justify.
5. The hands of a clock at $3$ o'clock form what angle? Is it a linear pair with the hand-positions at $9$ o'clock?
6. Two intersecting lines form angles $a, b, a, b$. If $a = 35^\circ$, find $b$.
7. Two adjacent angles together are $90^\circ$. What is each called?
8. Four rays meet at a point making angles $50^\circ, 70^\circ, 110^\circ, x$. Find $x$.

Activity

On a piece of paper, draw two lines crossing. With a protractor, measure all four angles. Verify that vertically opposite angles are equal and that each linear pair sums to $180^\circ$.