Question

lesson 15.1 homework
complete problems 1 - 7 for independent practice.
when you are finished, check the solutions with your teacher.
find the measure of the arc or central angle for the following circles.

1. find the measure of ∠rqs.

use the inscribed angle theorem to find missing angle measures.

1. find the measure of ∠d.
2. find the arc length of $overset{\frown}{bc}$.
3. find the measure of ∠d.

Explanation:

1. Problem 1:

• # Explanation:
• ## Step 1: Identify the given information
• There is no clear - given measure related to \(\angle RQS\) in the description. Assuming it is a central angle and if the given \(43^{\circ}\) is related to it (no clear indication in the provided text, but if it is the measure of the arc or related angle), and if \(\angle RQS\) is a central angle corresponding to an arc of measure \(43^{\circ}\), then \(\angle RQS = 43^{\circ}\) (central angle has the same measure as its intercepted arc).
• # Answer:
• \(43^{\circ}\)

1. Problem 2:

• # Explanation:
• ## Step 1: Recall the Inscribed - Angle Theorem
• The measure of an inscribed angle is half the measure of its intercepted arc. Let the intercepted arc of \(\angle D\) be \(124^{\circ}\).
• ## Step 2: Calculate the measure of \(\angle D\)
• By the Inscribed - Angle Theorem, \(\angle D=\frac{1}{2}\times124^{\circ}\).
• \(\angle D = 62^{\circ}\)
• # Answer:
• \(62^{\circ}\)

1. Problem 3:

• # Explanation:
• ## Step 1: Recall the Inscribed - Angle Theorem
• First, find the measure of the central angle corresponding to arc \(\overset{\frown}{BC}\). If the inscribed angle is \(56^{\circ}\), the central angle is \(2\times56^{\circ}=112^{\circ}\). But to find the arc length, we need the radius of the circle. Assuming the radius \(r\) is known, the formula for arc length \(s = r\theta\) (where \(\theta\) is in radians). Convert \(112^{\circ}\) to radians: \(\theta=\frac{112\pi}{180}=\frac{28\pi}{45}\). So \(s = r\times\frac{28\pi}{45}\). Since the radius is not given, if we just consider the angle measure of the arc, the measure of arc \(\overset{\frown}{BC}=112^{\circ}\).
• # Answer:
• \(112^{\circ}\) (assuming we are just looking for the angle - measure of the arc; if radius is given, arc length \(s = r\times\frac{28\pi}{45}\))

1. Problem 4:

• # Explanation:
• ## Step 1: Find the measure of the arc intercepted by \(\angle D\)
• The sum of the measures of the arcs of a circle is \(360^{\circ}\). Let the arc intercepted by \(\angle D\) be \(x\). We know two arcs are \(50^{\circ}\) and \(134^{\circ}\), so \(x=360-(50 + 134)=176^{\circ}\).
• ## Step 2: Use the Inscribed - Angle Theorem
• By the Inscribed - Angle Theorem, \(\angle D=\frac{1}{2}\times176^{\circ}\).
• \(\angle D = 88^{\circ}\)
• # Answer:
• \(88^{\circ}\)

Answer:

1. Problem 1:

• # Explanation:
• ## Step 1: Identify the given information
• There is no clear - given measure related to \(\angle RQS\) in the description. Assuming it is a central angle and if the given \(43^{\circ}\) is related to it (no clear indication in the provided text, but if it is the measure of the arc or related angle), and if \(\angle RQS\) is a central angle corresponding to an arc of measure \(43^{\circ}\), then \(\angle RQS = 43^{\circ}\) (central angle has the same measure as its intercepted arc).
• # Answer:
• \(43^{\circ}\)

1. Problem 2:

• # Explanation:
• ## Step 1: Recall the Inscribed - Angle Theorem
• The measure of an inscribed angle is half the measure of its intercepted arc. Let the intercepted arc of \(\angle D\) be \(124^{\circ}\).
• ## Step 2: Calculate the measure of \(\angle D\)
• By the Inscribed - Angle Theorem, \(\angle D=\frac{1}{2}\times124^{\circ}\).
• \(\angle D = 62^{\circ}\)
• # Answer:
• \(62^{\circ}\)

1. Problem 3:

• # Explanation:
• ## Step 1: Recall the Inscribed - Angle Theorem
• First, find the measure of the central angle corresponding to arc \(\overset{\frown}{BC}\). If the inscribed angle is \(56^{\circ}\), the central angle is \(2\times56^{\circ}=112^{\circ}\). But to find the arc length, we need the radius of the circle. Assuming the radius \(r\) is known, the formula for arc length \(s = r\theta\) (where \(\theta\) is in radians). Convert \(112^{\circ}\) to radians: \(\theta=\frac{112\pi}{180}=\frac{28\pi}{45}\). So \(s = r\times\frac{28\pi}{45}\). Since the radius is not given, if we just consider the angle measure of the arc, the measure of arc \(\overset{\frown}{BC}=112^{\circ}\).
• # Answer:
• \(112^{\circ}\) (assuming we are just looking for the angle - measure of the arc; if radius is given, arc length \(s = r\times\frac{28\pi}{45}\))

1. Problem 4:

• # Explanation:
• ## Step 1: Find the measure of the arc intercepted by \(\angle D\)
• The sum of the measures of the arcs of a circle is \(360^{\circ}\). Let the arc intercepted by \(\angle D\) be \(x\). We know two arcs are \(50^{\circ}\) and \(134^{\circ}\), so \(x=360-(50 + 134)=176^{\circ}\).
• ## Step 2: Use the Inscribed - Angle Theorem
• By the Inscribed - Angle Theorem, \(\angle D=\frac{1}{2}\times176^{\circ}\).
• \(\angle D = 88^{\circ}\)
• # Answer:
• \(88^{\circ}\)