https://file.aiquickdraw.com/mathbot/file/17271921585430d29pen4.jpg

To find the measure of \(\angle QTN\), we can use the information provided in the image.

### Step 1: Use the given angles.
We are given two expressions for angles at points \(S\) and \(T\):

- At \(S\), the angle is \(2x + 55^\circ\).
- At \(T\), the angle is \(7x + 25^\circ\).

These two angles are vertically opposite angles, and vertically opposite angles are equal. Therefore, we can set these two expressions equal to each other:

\[
2x + 55^\circ = 7x + 25^\circ
\]

### Step 2: Solve for \(x\).
Now we will solve the equation to find \(x\):

\[
2x + 55 = 7x + 25
\]

Subtract \(2x\) from both sides:

\[
55 = 5x + 25
\]

Subtract 25 from both sides:

\[
30 = 5x
\]

Divide by 5:

\[
x = 6
\]

### Step 3: Find the measure of \(\angle QTN\).
Now that we know \(x = 6\), substitute this value into the expression for \(\angle QTN\), which is \(7x + 25^\circ\):

\[
\angle QTN = 7(6) + 25 = 42 + 25 = 67^\circ
\]

Thus, the measure of \(\angle QTN\) is \(67^\circ\).

Would you like further explanation or have any questions?

### Related Questions:
1. How are vertically opposite angles used in solving geometry problems?
2. Can we use supplementary angles in this scenario? Why or why not?
3. What are some other ways to verify if two lines are parallel using angles?
4. How can we find the measure of other angles in the figure?
5. How would the problem change if the angles were given differently?

### Tip:
When solving problems involving angles in intersecting lines, always check for relationships like vertically opposite angles, corresponding angles, or alternate interior angles.

Find the measure of ∠QTN based on the angles provided at points S and T in the figure.

Find the measure of angle QTN using vertically opposite angles. This problem involves solving the equation 2x + 55 = 7x + 25 to find the value of x and then substituting it to calculate ∠QTN. Suitable for Grade 9-10 students.

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