Solve for x.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**Solve for \( x \).**
The given problem features a circle with center \( T \) and various marked points on the circumference, identified as \( Q \), \( R \), and \( S \). Here's a step-by-step explanation of the diagram:
1. **Angle \( \angle QPR \) Inside the Circle:**
The angle \( \angle QPR \) measures \( 67^\circ \).
2. **Angle \( \angle QTS \) Outside the Circle:**
The angle \( \angle QTS \) measures \( 120^\circ \).
3. **Angle \( \angle QRS \):**
The angle \( \angle QRS \) is labeled as \( (5x + 21)^\circ \).
To solve for \( x \):
1. **Understand Circle Properties:**
Since point \( T \) is the center of the circle and \( PR \) is a chord, we should recognize that the external angle \( \angle QTS = 120^\circ \) subtends both \( \angle QSR \) and \( \angle SQR \).
2. **Using the Angle Sum Property in a Circle:**
We know that the angle subtended by the same arc on a circle is equal. Therefore, the entire subtended angle around the circle sums up considering all prominent segments leading around \( T \). These include \( 67^\circ\), \( 120^\circ\), and the target angle, \( (5x + 21) \).
Upon correctly summing and equating factors, proper calculation transformation or assumption is required to directly solve \( x \) by understanding:
Formulate appropriate segments consideration if based on basic angle properties:
\[ \angle QRS = \frac{120^\circ + 67^\circ}{2} \]
Equating expression \(5x + 21 = 93.5 - 21\):
\( 5x \) simplifies or produces through steps:
\( x = 3.2 \)
(Note: Further solutions model authenticity reflects exact practices revisiting interpret under Euclidean dynamic apply).
Hence, value of \( x \) simplified:
\[ x = 14.5^\circ \ via align assumptions\ or Next validation further]; Otherwise transformed and corrected) accurate identify steps.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc9a89eb-87cc-4d60-b665-4a120d0798e8%2F9edeaa74-596f-4169-81b7-03220cf69ec2%2Fxb92vj_processed.jpeg&w=3840&q=75)
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