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.
![---
**Finding the Degree Measure of Arc AW**
**Diagram Explanation:**
- The diagram is a circle divided by lines into four arcs: \( AW, WC, CB, \) and \( BA \).
- \( W \) is the point at the top of the circle.
- \( A \) is the point on the left of the circle.
- \( C \) is the point on the right of the circle.
- \( B \) is the point at the bottom of the circle.
- \( D \) is the center of the circle, and the angle at \( D \) inside the circle connected to \( CB \) is given as \( 85^\circ \).
**Given Information:**
- Arc \( AW \) is represented by the expression \( 6x + 16 \) degrees.
- Arc \( CB \) is represented by the expression \( 10 + 10x \) degrees.
**Problem Statement:**
You need to find the degree measure of arc \( AW \).
**Solution Steps:**
1. Understand that the sum of angles around point \( D \) in a circle is always \( 360^\circ \).
2. Set up the equation including all arc measures around the circle:
\[
6x + 16 + 85 + (10 + 10x) = 360
\]
3. Combine like terms:
\[
6x + 10x + 16 + 85 + 10 = 360
\]
\[
16x + 111 = 360
\]
4. Solve for \( x \):
\[
16x = 360 - 111
\]
\[
16x = 249
\]
\[
x = \frac{249}{16}
\]
\[
x = 15.5625
\]
5. Substitute \( x \) back into the expression for arc \( AW \):
\[
AW = 6x + 16
\]
\[
AW = 6(15.5625) + 16
\]
\[
AW = 93.375 + 16
\]
\[
AW = 109.375
\]
**Conclusion:**
The degree measure of arc \( AW \) is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03ee4b44-fd9b-4d43-bc23-25b15a4cdf81%2F631d497e-d304-43d0-9d79-f1d9323f3c17%2Fqg61dw_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
