QS bisects angle PQT, the measurement of angle SQT= 8x-25, the measurement of angle PQT= 9x + 34,and the measurement of angle SQR = 112, find each measure

Let's solve the problem step by step.

### Given:
- QS bisects \( \angle PQT \), which means \( \angle PQS = \angle SQT \).
- \( m \angle SQT = 8x - 25 \)
- \( m \angle PQT = 9x + 34 \)
- \( m \angle SQR = 112^\circ \)

### Step 1: Use the bisecting property
Since QS bisects \( \angle PQT \), we have:
\[
m \angle PQS = m \angle SQT
\]
But also:
\[
m \angle PQT = m \angle PQS + m \angle SQT
\]

Given that \( m \angle PQS = m \angle SQT \), we can express:
\[
m \angle PQT = 2 \times m \angle SQT
\]

Substituting the given expressions:
\[
9x + 34 = 2 \times (8x - 25)
\]

### Step 2: Solve for \( x \)
Expand and simplify the equation:
\[
9x + 34 = 16x - 50
\]

Bring all terms involving \( x \) to one side and constants to the other:
\[
34 + 50 = 16x - 9x
\]

\[
84 = 7x
\]

\[
x = \frac{84}{7} = 12
\]

### Step 3: Calculate the angle measurements
Now that we have \( x = 12 \), we can find the actual measurements of the angles.

1. **\( m \angle SQT \)**:
   \[
   m \angle SQT = 8x - 25 = 8(12) - 25 = 96 - 25 = 71^\circ
   \]

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

3. **\( m \angle PQS \)**:
   Since QS bisects \( \angle PQT \), \( m \angle PQS = m \angle SQT = 71^\circ \).

4. **\( m \angle SQR \)**:
   Given directly as \( m \angle SQR = 112^\circ \).

### Final Measurements:
- \( m \angle SQT = 71^\circ \)
- \( m \angle PQT = 142^\circ \)
- \( m \angle PQS = 71^\circ \) (since it’s equal to \( m \angle SQT \))
- \( m \angle SQR = 112^\circ \)

Would you like further explanation on any part of this solution, or do you have additional questions?

Here are five related questions you might find interesting:

1. How do you determine if two angles are congruent?
2. What is the relationship between the exterior and interior angles of a triangle?
3. How do you find the unknown angles in an isosceles triangle?
4. What is the significance of angle bisectors in a triangle?
5. How can the Angle Sum Property of a triangle help solve angle-related problems?

**Tip:** When working with bisected angles, remember that the bisector divides the angle into two equal parts, which can simplify the calculations significantly.

Explore how to solve a geometry problem involving angle bisectors. Given that QS bisects angle PQT, with specific angle measurements provided, learn how to calculate each angle using the bisecting property. Suitable for students in Grades 9-10.

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