Let's go through the three problems presented in the image step by step:

Problem 13

If $m\angle 1 = 78^\circ$, what is the $m\angle 2$ if the two angles are complementary?

• Complementary angles sum up to $90^\circ$. Therefore, to find $m\angle 2$: $m\angle 2 = 90^\circ - 78^\circ = 12^\circ$

  Answer for Problem 13: $m\angle 2 = 12^\circ$

Problem 14

If $\overline{BD}$ bisects $\angle ABC$, solve for $x$.

• The given angle information:

  • $\angle ABD = 30^\circ$
  • $\angle DBC = (x + 25)^\circ$

• Since $\overline{BD}$ bisects $\angle ABC$, the two angles $\angle ABD$ and $\angle DBC$ must be equal: $30^\circ = (x + 25)^\circ$ Solving for $x$: $x = 30^\circ - 25^\circ = 5^\circ$

  Answer for Problem 14: $x = 5^\circ$

Problem 15

Solve for $x$ and $y$.

• The given system involves two vertical angles, which are equal:

  • $5x = 12x - 41$
  • $5y = 5y$

  From the first equation, solve for $x$: $5x = 12x - 41$ $41 = 12x - 5x$ $41 = 7x \quad \Rightarrow \quad x = \frac{41}{7} \approx 5.86$

  For $y$, since both equations involving $y$ are the same, $y$ is not uniquely determined by the problem, and the value for $y$ remains arbitrary.

  Answer for Problem 15: $x \approx 5.86$, $y$ is arbitrary.

Would you like more details on any of these steps?

Here are five related questions for further practice:

1. What are supplementary angles, and how do you calculate them?
2. How would you solve for a different variable in a bisected angle problem?
3. What happens when two angles are both complementary and bisected?
4. How can you check if a given system of linear equations has a unique solution?
5. How would you interpret a system of equations where one variable is not determined?

Tip: Always label your angle pairs and write out the equations based on their relationships (complementary, supplementary, bisected) for clarity!