if QS bisects angle PQT, angle SQT = (8x-25), angle PQT = (9x+34), and angle SQR = 112, find each measure.

Given that \( QS \) bisects \( \angle PQT \), let's break down the problem:

- \( \angle PQT = 9x + 34 \)
- Since \( QS \) bisects \( \angle PQT \), the two angles formed, \( \angle PQS \) and \( \angle SQT \), are equal.
- So, \( \angle PQS = \angle SQT = \frac{1}{2} \times \angle PQT \).
- \( \angle SQT = 8x - 25 \) (given)
- \( \angle SQR = 112^\circ \) (given)

Let's find \( x \) first by setting \( \angle SQT = \frac{1}{2} \times \angle PQT \):

\[
8x - 25 = \frac{1}{2} \times (9x + 34)
\]

Now, solve for \( x \):

\[
8x - 25 = \frac{9x + 34}{2}
\]

Multiply both sides by 2 to eliminate the fraction:

\[
16x - 50 = 9x + 34
\]

Now, solve for \( x \):

\[
16x - 9x = 34 + 50
\]
\[
7x = 84
\]
\[
x = 12
\]

Now that we have \( x = 12 \), we can find the measures of the angles:

1. **\( \angle PQT \)**:
   \[
   \angle PQT = 9x + 34 = 9(12) + 34 = 108 + 34 = 142^\circ
   \]

2. **\( \angle SQT \)** (which is also \( \angle PQS \) because \( QS \) bisects \( \angle PQT \)):
   \[
   \angle SQT = 8x - 25 = 8(12) - 25 = 96 - 25 = 71^\circ
   \]

3. **\( \angle SQR \)** is given as \( 112^\circ \).

Thus, the angles are:
- \( \angle PQT = 142^\circ \)
- \( \angle SQT = \angle PQS = 71^\circ \)
- \( \angle SQR = 112^\circ \) (already given)

Would you like any further clarification on these steps?

Here are 5 related questions:
1. What if \( \angle SQT \) was given instead as \( 7x - 15 \)? How would the angles change?
2. How can you use the sum of angles in a triangle to verify the calculation?
3. If \( \angle SQR \) was changed, how would that affect the overall solution?
4. What if \( QS \) didn't bisect \( \angle PQT \), but instead was given as another angle?
5. Can you solve a similar problem if an exterior angle and one interior angle are given?

**Tip:** When working with angle bisectors, always remember that the two resulting angles are equal, which often leads to setting up a helpful equation.

Explore how to apply the angle bisector theorem to find angle measures in geometry. Solve for angles PQT, SQT, and SQR using given angle relationships and the angle bisector theorem. Suitable for Grades 9-12.

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