In quadrilateral QRST, m∠Q > is 68°, m∠R is (3x + 40)°, and m∠T is (5x − 52)°. What are the measures of ∠R, ∠S, and ∠T? Write the numerical values in that order with the measures separated by commas.  

     
quadrilateral QRST inscribed in circle P

Since quadrilateral \( QRST \) is inscribed in a circle, it is a cyclic quadrilateral. This implies that opposite angles are supplementary, meaning their measures add up to \( 180^\circ \).

Given:
- \( m\angle Q = 68^\circ \)
- \( m\angle R = (3x + 40)^\circ \)
- \( m\angle T = (5x - 52)^\circ \)

Since \( \angle Q \) and \( \angle S \) are opposite angles, they should add up to \( 180^\circ \):
\[
m\angle Q + m\angle S = 180^\circ
\]
Thus,
\[
68^\circ + m\angle S = 180^\circ
\]
\[
m\angle S = 180^\circ - 68^\circ = 112^\circ
\]

Now, \( \angle R \) and \( \angle T \) are also opposite angles and should add up to \( 180^\circ \):
\[
(3x + 40) + (5x - 52) = 180
\]
Simplifying this equation:
\[
8x - 12 = 180
\]
\[
8x = 192
\]
\[
x = 24
\]

Now, substitute \( x = 24 \) into the expressions for \( m\angle R \) and \( m\angle T \):
\[
m\angle R = 3(24) + 40 = 72 + 40 = 112^\circ
\]
\[
m\angle T = 5(24) - 52 = 120 - 52 = 68^\circ
\]

Therefore, the measures are:
\[
m\angle R = 112^\circ, \; m\angle S = 112^\circ, \; m\angle T = 68^\circ
\]

Would you like to see the step-by-step process in more detail?

1. How does the cyclic property of quadrilaterals affect angle relationships?
2. How can we verify if a quadrilateral is cyclic?
3. What are some real-world applications of cyclic quadrilaterals?
4. Could this approach be applied to non-cyclic quadrilaterals?
5. Why is it important that \( QRST \) is inscribed in circle \( P \)?

**Tip:** When working with cyclic quadrilaterals, remember that opposite angles will always add up to \( 180^\circ \).

In quadrilateral QRST, m∠Q > is 68°, m∠R is (3x + 40)°, and m∠T is (5x − 52)°. What are the measures of ∠R, ∠S, and ∠T? Write the numerical values in that order with the measures separated by commas.

Learn how to solve for the measures of angles ∠R, ∠S, and ∠T in a cyclic quadrilateral QRST, given that m∠Q = 68°, m∠R = (3x + 40)°, and m∠T = (5x − 52)°. The solution uses the properties of cyclic quadrilaterals and algebraic equations.

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