Assignment Practice with inscribed angle theorems and corollaries and the angle formed by a tangent and chord theorem Q 84° R S What is the measure of QS?
The Correct Answer and Explanation is:
Correct Answer: 168
Explanation:
To determine the measure of arc QS, we need to use the inscribed angle theorem. This fundamental theorem in geometry establishes a relationship between an inscribed angle and its intercepted arc.
An inscribed angle is an angle formed by two chords in a circle that have a common endpoint on the circle. In the given image, the angle ∠QRS is an inscribed angle because its vertex, R, is on the circle, and its sides, QR and SR, are chords of the circle.
The intercepted arc is the portion of the circle that lies in the interior of the inscribed angle. For the inscribed angle ∠QRS, the intercepted arc is arc QS. This is the part of the circle’s circumference that is “cut off” by the sides of the angle.
The inscribed angle theorem states that the measure of an inscribed angle is exactly half the measure of its intercepted arc. We can express this relationship with the formula:
Measure of Inscribed Angle = (1/2) × Measure of Intercepted Arc
In this problem, we are given the measure of the inscribed angle ∠QRS, which is 84°. We need to find the measure of its intercepted arc, arc QS.
Using the formula, we can set up the equation:
m∠QRS = (1/2) × m(arc QS)
Now, we substitute the known value into the equation:
84° = (1/2) × m(arc QS)
To solve for the measure of arc QS, we need to isolate it. We can do this by multiplying both sides of the equation by 2:
2 × 84° = m(arc QS)
Performing the multiplication gives us the final answer:
168° = m(arc QS)
Therefore, the measure of arc QS is 168 degrees.
