Question: A beam of light and a mirror can be used to study the behavior of light. Light that strikes the mirror is reflected so that the angle of reflection and the angle of incidence are congruent. In the diagram, $\angle ABC$ has a measure of $39^\circ$. a. Name the angle of reflection and find its measure. b. Find $m\angle ABD$. c. Find $m\angle ABE$ and $m\angle DBF$. Angle of incidence Angle of reflection a. Name the angle of reflection and find its measure: The angle of reflection is $\angle B$ and has a measure of $\square^\circ$.

1. **State the problem:** We are given a beam of light striking a mirror at point $B$. The angle of incidence $\angle ABC$ is $39^\circ$. We need to find the angle of reflection and other related angles. 2. **Recall the law of reflection:** The angle of incidence equals the angle of reflection. That means: $$m\angle ABC = m\angle ABD = 39^\circ$$ where $\angle ABD$ is the angle of reflection. 3. **Find the angle of reflection:** Since $m\angle ABC = 39^\circ$, the angle of reflection $m\angle ABD$ is also $39^\circ$. 4. **Find $m\angle ABD$:** From step 3, $m\angle ABD = 39^\circ$. 5. **Find $m\angle ABE$ and $m\angle DBF$:** - $\angle ABE$ is the angle between the incident ray $AB$ and the mirror line $BE$. Since the mirror line is horizontal and $\angle ABC$ is $39^\circ$ from the normal $BC$ (which is vertical), the angle between the incident ray and the mirror is complementary to $39^\circ$: $$m\angle ABE = 90^\circ - 39^\circ = 51^\circ$$ - Similarly, $\angle DBF$ is the angle between the reflected ray $BD$ and the mirror line $BF$. It is also complementary to the angle of reflection: $$m\angle DBF = 90^\circ - 39^\circ = 51^\circ$$ **Final answers:** - Angle of reflection $\angle ABD = 39^\circ$ - $m\angle ABD = 39^\circ$ - $m\angle ABE = 51^\circ$ - $m\angle DBF = 51^\circ$