Angles basics

Most angle questions are solved by combining a small set of facts, one step at a time. Around a single point the angles make a full turn, so they add to $360^\circ$. Angles that sit on a straight line are supplementary, adding to $180^\circ$. And when two lines cross, the angles directly across from each other, called vertically opposite angles, are equal.

In this figure of two crossing lines, the angles marked $a$ are vertically opposite and so are equal, as are the two marked $b$; since an $a$ and a neighbouring $b$ lie together on a straight line, $a + b = 180^\circ$.

When a transversal crosses two parallel lines, eight angles are formed, numbered 1 to 8 in the figure below.

Three relationships unlock all of them. Corresponding angles occupy the same position at the two crossings and are equal, for example angle 4 and angle 8. Alternate angles lie on opposite sides of the transversal and between the two lines, and they are equal too, for example angle 4 and angle 6. Co-interior (same-side) angles sit between the lines on the same side of the transversal and are supplementary, adding to $180^\circ$, for example angle 4 and angle 5.

The skill is chaining these facts: find one angle, use it to get the next, and continue. For instance, a vertically opposite angle gives you one value, a corresponding angle carries that value to a second angle, and a supplementary pair then gives a third. Each step uses just one fact.

Two lines cross so that one angle is $110^\circ$. Its vertically opposite angle is also $110^\circ$, and each angle next to it on the straight line is $180^\circ - 110^\circ = 70^\circ$.

A transversal crosses two parallel lines. If one angle is $97^\circ$, the corresponding angle on the other parallel line is also $97^\circ$, and the co-interior angle on the same side is $180^\circ - 97^\circ = 83^\circ$.

Two co-interior angles are $(3x + 10)^\circ$ and $(2x + 20)^\circ$. Since they are supplementary, $(3x + 10) + (2x + 20) = 180$, so $5x + 30 = 180$, giving $x = 30$.