Question

find the measure of the missing angles.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. The angle opposite the $92^{\circ}$ angle has a measure of $92^{\circ}$. Let's find angle $d$. The sum of angles around a point is $360^{\circ}$. Consider the non - overlapping angles formed at the intersection. We know one angle is $92^{\circ}$ and another is $21^{\circ}$. Let's first find the angle adjacent to the $21^{\circ}$ angle that forms a linear pair with $d$.

Step2: Find the adjacent angle to $21^{\circ}$

The angle adjacent to the $21^{\circ}$ angle and the $92^{\circ}$ angle forms a linear pair with $d$. The sum of angles on a straight - line is $180^{\circ}$. Let the angle adjacent to $21^{\circ}$ be $x$. Then $x=180^{\circ}-(92^{\circ} + 21^{\circ})=180^{\circ}-113^{\circ}=67^{\circ}$.

Step3: Find angle $d$

Since $d$ and the $67^{\circ}$ angle are vertical angles, $d = 67^{\circ}$.

Step4: Find angle $f$

Angle $f$ and the $92^{\circ}$ angle are vertical angles, so $f = 92^{\circ}$.

Step5: Find angle $e$

Angle $e$ and the $21^{\circ}$ angle are vertical angles, so $e = 21^{\circ}$.

Answer:

$d = 67^{\circ}$, $e = 21^{\circ}$, $f = 92^{\circ}$