Angles and transversals

Step 1: Look at one intersection

If we focus on the top cluster of angles:

• $\angle A$ and $\angle D$ are opposite angles (also called vertical angles), so they are congruent.
• $\angle B$ and $\angle C$ are also opposite angles, so they are congruent.

Now look at pairs that form a straight line:

• $\angle A$ and $\angle B$ form a straight line, so they are supplementary angles.
• The same is true for $\angle B$ and $\angle D$, $\angle D$ and $\angle C$, and $\angle C$ and $\angle A$.

Step 2: Connect both intersections

Because the two horizontal lines are parallel, the same angle pattern repeats at both intersections: every acute angle (green in the image) is congruent to every other acute angle, and every obtuse angle (gold) is congruent to every other obtuse angle. Any acute angle and any obtuse angle together are supplementary, since they add up to $180^{\circ }$. This shortcut works only when the lines are parallel; if they aren’t, only vertical angles and straight-line pairs are guaranteed.

Two of these cross-intersection pairs have names you’ll see on the test:

• Corresponding angles sit in the same position at each intersection, like $\angle A$ and $\angle E$. They’re congruent.
• Alternate interior angles sit between the parallel lines, on opposite sides of the transversal, like $\angle C$ and $\angle F$ (or $\angle D$ and $\angle E$). They’re also congruent.

Step 3: Apply the pattern

Using the rule above, you can label every angle in the figure just by knowing whether it’s acute or obtuse. For example, $\angle D$ (gold/obtuse) and $\angle F$ (green/acute) are supplementary. $\angle F$ and $\angle G$ are opposite angles, so they’re congruent. And $\angle F$ and $\angle E$ form a straight line, so they’re supplementary too - which means $\angle E$ is gold, matching $\angle A$, $\angle D$, and $\angle H$.

Example: Angles across a transversal

In the figure above, lines $A$ and $B$ are parallel. If $\angle X$ is $55^{\circ }$, what is the measure of $\angle S$?

This is asking: how can we connect $\angle S$ to the angle we know?

• $\angle X$ and $\angle U$ are alternate interior angles (both acute, between the parallel lines, on opposite sides of the transversal), so they are congruent: $\angle U=55^{\circ }$
• $\angle U$ and $\angle S$ form a straight line, so they are supplementary

$$\begin{array}{rl}S+U& =180\\ S+55& =180\\ S& =125\end{array}$$

$\angle S=125^{\circ }$