Let's go through the questions you provided in the image one by one.

Problem 12:

Question:
"The measure of the supplement of an angle exceeds 3 times the measure of the complement of the angle by 10. Find the measure of the complement."

Solution:
Let $x$ be the measure of the angle.

• The supplement of the angle is $180^\circ - x$.
• The complement of the angle is $90^\circ - x$.

According to the problem:
$180^\circ - x = 3(90^\circ - x) + 10$

Expand and solve for $x$:
$180^\circ - x = 270^\circ - 3x + 10$ $- x + 3x = 270^\circ + 10 - 180^\circ$ $2x = 100^\circ$ $x = 50^\circ$

Thus, the complement of the angle is $90^\circ - 50^\circ = 40^\circ$.

Problem 15:

Question:
Given $PQ \perp QR$, find $m \angle PQS$.

Solution:
Since $PQ$ is perpendicular to $QR$, $\angle PQS$ is part of a right angle.

Given that:
$m \angle PQS = x + 12^\circ$ $m \angle QRS = 43^\circ - 2x$

Using the fact that angles around point Q sum to $90^\circ$ and substituting the given expressions will allow you to solve for $x$ and find the measure of $\angle PQS$.

Problem 16:

Question:
Given:

• $\angle 1$ is complementary to $\angle 4$.
• $\angle 2$ is complementary to $\angle 3$.
• $\overline{RT}$ bisects $\angle SRV$.

Prove that $\overline{TR}$ bisects $\angle STV$.

Proof Outline:

1. Complementary Angles Relationship:

   • Since $\angle 1$ is complementary to $\angle 4$ and $\angle 2$ is complementary to $\angle 3$, we have $m\angle 1 + m\angle 4 = 90^\circ$ and $m\angle 2 + m\angle 3 = 90^\circ$.

2. Bisected Angles:

   • $\overline{RT}$ bisects $\angle SRV$, which means $m\angle 3 = m\angle 4$.

3. Applying the Complementary Relationship:

   • Since $\angle 1 + \angle 4 = 90^\circ$ and $\angle 2 + \angle 3 = 90^\circ$, and $\angle 3 = \angle 4$, it follows that $\angle 1 = \angle 2$.

4. Conclusion:

   • Since $\angle 1 = \angle 2$, and $\angle 1$ is on the left side of the bisected angle and $\angle 2$ is on the right side, $\overline{TR}$ must also bisect $\angle STV$.

Would you like a more detailed explanation or help with any of the steps?

Here are 5 follow-up questions you might consider:

1. What is the relationship between complementary and supplementary angles in terms of their sums?
2. How does the perpendicularity condition impact the angles around a point?
3. What is the significance of angle bisectors in geometric proofs?
4. How can you use angle relationships to solve for unknown variables?
5. How do the properties of complementary angles help in proving congruency in geometric figures?

Tip: When solving geometry problems, always clearly define the relationships between given angles, especially when dealing with complementary or supplementary angles.