Angles and transversals

Now let’s connect the top cluster to the bottom cluster.

If we draw a second blue line (a transversal) parallel to the first transversal, we create a parallelogram with new angles $\angle I$ and $\angle J$.

Step 3: Follow the relationships

Now we use what we know to trace relationships across both intersections. That means $\angle D$ and $\angle F$ are supplementary.

We already know $\angle D$ and $\angle C$ are supplementary, so $\angle C$ and $\angle F$ must be congruent.

From there:

• $\angle G$ is congruent to $\angle F$ because they are opposite angles
• $\angle F$ and $\angle E$ are supplementary because they form a straight line

In the image, green angles are supplementary with gold angles.

That means $\angle E$ must match the gold angles, and $\angle H$ (opposite and therefore congruent with $\angle E$) must also match.

We have now accounted for the relationships between all of the angles.

Practice problem

Example:

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 congruent, so $\angle U=55^{\circ }$
• $\angle U$ and $\angle S$ form a straight line, so they are supplementary

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

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

Steps taken:

• Find a congruent angle
• Use a supplementary relationship
• Solve for the unknown