In ΔQRS, m, angle, Q, equals, left parenthesis, 6, x, plus, 17, right parenthesis, degreesm∠*Q*=(6*x*+17)∘, m, angle, R, equals, left parenthesis, x, plus, 5, right parenthesis, degreesm∠*R*=(*x*+5)∘, and m, angle, S, equals, left parenthesis, 3, x, minus, 2, right parenthesis, degreesm∠*S*=(3*x*−2)∘. Find m, angle, R, .m∠*R*.

To solve for \( m\angle R \), we need to find the value of \( x \) first by using the fact that the sum of the angles in any triangle is always \( 180^\circ \).

Given:
- \( m\angle Q = (6x + 17)^\circ \)
- \( m\angle R = (x + 5)^\circ \)
- \( m\angle S = (3x - 2)^\circ \)

The sum of the angles in triangle \( \Delta QRS \) is:

\[
m\angle Q + m\angle R + m\angle S = 180^\circ
\]

Substitute the given expressions:

\[
(6x + 17) + (x + 5) + (3x - 2) = 180^\circ
\]

Now, combine like terms:

\[
6x + x + 3x + 17 + 5 - 2 = 180
\]
\[
10x + 20 = 180
\]

Subtract 20 from both sides:

\[
10x = 160
\]

Divide by 10:

\[
x = 16
\]

Now that we have \( x = 16 \), we can substitute this value back into the expression for \( m\angle R \):

\[
m\angle R = x + 5 = 16 + 5 = 21^\circ
\]

So, the measure of \( m\angle R \) is \( 21^\circ \).

Would you like more details or have any questions?

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Here are five related questions:
1. What is the measure of \( m\angle Q \) after finding \( x \)?
2. What is the measure of \( m\angle S \) after finding \( x \)?
3. How would you verify the sum of the angles in \( \Delta QRS \)?
4. What if \( m\angle Q \) was known—how would that change the approach?
5. How does knowing \( x \) help in solving for other unknowns in the triangle?

**Tip:** Always substitute back into all original expressions to verify your solution is consistent with all given information.

Learn how to find the measure of angle R in triangle QRS using the angle sum property of triangles and solving linear equations. Follow a step-by-step solution to determine the angle R with detailed calculations. Suitable for high school students.

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