Question 174·Medium·Geometry and Trigonometry

Think about how angle $x$ at $E$ is made by the two diagonals.

Draw (or imagine) a line through $E$ parallel to the bases. The angles along that straight line add to $180^\circ$.

In Desmos, type a=35 and b=25.

Type x=180-a-b.

Use the value displayed for x as the measure of angle $x$.

Since $\overline{AB} \parallel \overline{CD}$:

• Diagonal $\overline{AC}$ is a transversal, so the angle it makes with $\overline{AB}$ equals the labeled $35^\circ$ angle it makes with $\overline{CD}$.

• Diagonal $\overline{BD}$ is a transversal, so the angle it makes with $\overline{AB}$ equals the labeled $25^\circ$ angle it makes with $\overline{CD}$.

Imagine a line through $E$ that is parallel to $\overline{AB}$ (and therefore also parallel to $\overline{CD}$).

Above that line at $E$, the ray $\overline{EA}$ makes a $35^\circ$ angle with the line, and the ray $\overline{EB}$ makes a $25^\circ$ angle with the line.

Those two angles and angle $x$ fill a straight angle (a linear pair), so their measures add to $180^\circ$.

Therefore, the measure of angle $x$ is $120^\circ$.