In ΔQRS,  start overline, Q, S, end overline
QS
​
is extended through point S to point T,  m, angle, R, S, T, equals, left parenthesis, 6, x, plus, 7, right parenthesis, degreesm∠RST=(6x+7)
∘
,  m, angle, Q, R, S, equals, left parenthesis, x, minus, 3, right parenthesis, degreesm∠QRS=(x−3)
∘
,   and m, angle, S, Q, R, equals, left parenthesis, 3, x, plus, 20, right parenthesis, degreesm∠SQR=(3x+20)
∘
.   Find m, angle, S, Q, R, .m∠SQR.

To solve the problem, we need to find the measure of angle \( m\angle SQR \). We are given the following information about the angles in triangle \( \Delta QRS \) and the extended line segment:

- \( m\angle RST = (6x + 7)^\circ \)
- \( m\angle QRS = (x - 3)^\circ \)
- \( m\angle SQR = (3x + 20)^\circ \)

In any triangle, the sum of the interior angles is always \(180^\circ\). Therefore, in triangle \( \Delta QRS \), we have:

\[
m\angle QRS + m\angle SQR + m\angle QSR = 180^\circ
\]

But we are not given \( m\angle QSR \) directly. However, \( m\angle RST \) is the exterior angle at point \( S \) which is equal to the sum of the two opposite interior angles, \( m\angle SQR \) and \( m\angle QRS \):

\[
m\angle RST = m\angle SQR + m\angle QRS
\]

Substitute the given expressions:

\[
6x + 7 = (3x + 20) + (x - 3)
\]

Now, let's solve for \( x \):

\[
6x + 7 = 3x + x + 20 - 3
\]
\[
6x + 7 = 4x + 17
\]
\[
6x - 4x = 17 - 7
\]
\[
2x = 10
\]
\[
x = 5
\]

Now that we have \( x = 5 \), substitute this value back into the expression for \( m\angle SQR \):

\[
m\angle SQR = 3x + 20 = 3(5) + 20 = 15 + 20 = 35^\circ
\]

So, the measure of \( m\angle SQR \) is \( 35^\circ \).

Would you like more details or have any questions?

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Here are five related questions:
1. What is the measure of angle \( m\angle QRS \)?
2. What is the measure of angle \( m\angle RST \)?
3. What is the sum of the angles in triangle \( \Delta QRS \)?
4. How does the exterior angle theorem help in solving this problem?
5. Can you verify the solution by finding \( m\angle QSR \)?

**Tip:** Always check the relationships between interior and exterior angles when working with triangles extended to form exterior angles.

Learn how to solve a triangle geometry problem involving angles in triangle QRS and the exterior angle theorem. Find the measure of angle SQR with step-by-step explanation and calculations. Suitable for high school students.

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