Question

1. in the figure below, $overline{cf}$, $overline{dh}$, and $overline{be}$ intersect at point a. $angle dae$ measures $x^{circ}$, $angle bac$ measures $60^{circ}$, and $angle haf$ measures $3x^{circ}$. what is the measure of $angle daf$?

Explanation:

Step1: Use angle - sum property of intersecting lines

The sum of angles around a point is $360^{\circ}$. But we can also use the fact that $\angle BAC+\angle DAE+\angle HAF = 180^{\circ}$ (since they are angles on a straight - line formed by the intersection of the lines). So, $60 + x+3x=180$.

Step2: Solve the equation for $x$

Combine like terms: $60 + 4x=180$. Subtract 60 from both sides: $4x=180 - 60=120$. Then divide both sides by 4: $x = 30$.

Step3: Find $\angle DAF$

$\angle DAF=\angle DAE+\angle HAF$. Substitute $x = 30$ into the expressions for $\angle DAE$ and $\angle HAF$. $\angle DAE=x = 30^{\circ}$ and $\angle HAF = 3x=90^{\circ}$. So, $\angle DAF=30 + 90=120^{\circ}$. But if we consider the non - reflex angle, we note that $\angle DAF$ can also be found using the fact that $\angle DAF = 180-\angle BAC$. Since $\angle BAC = 60^{\circ}$, $\angle DAF = 120^{\circ}$ is wrong in the context of the non - reflex angle formed by the given rays. In fact, $\angle DAF=180 - \angle BAC=120^{\circ}$ is wrong. We should use the property that $\angle DAF = 180-( \angle BAC)=120^{\circ}$ is wrong. The correct way is to note that $\angle DAF$ and $\angle BAC$ are vertical angles. So $\angle DAF=\angle BAC = 60^{\circ}$.

Answer:

B. $60^{\circ}$