Question

if \\(m\widehat{vs} = 180^{\circ}\\) and \\(m\angle z = 58^{\circ}\\), find \\(m\widehat{wa}\\).

Explanation:

Identify the geometric relationship

The angle \(\angle Z\) is formed by two secants intersecting outside the circle. According to the Secant-Secant Angle Theorem, the measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of their intercepted arcs.

Set up the equation

Let the intercepted arcs be the far arc \(\widehat{VS}\) and the near arc \(\widehat{WA}\).
The formula is:

$$m\angle Z = \frac{1}{2}(m\widehat{VS} - m\widehat{WA})$$

Substitute the given values

We are given:

• \(m\widehat{VS} = 180^\circ\)
• \(m\angle Z = 58^\circ\)

Substitute these values into our equation:

$$58^\circ = \frac{1}{2}(180^\circ - m\widehat{WA})$$

Solve for the unknown arc measure

Using Linear Equations:
Multiply both sides by 2:

$$116^\circ = 180^\circ - m\widehat{WA}$$

Subtract \(180^\circ\) from both sides:

$$-64^\circ = -m\widehat{WA}$$

Multiply by \(-1\):

$$m\widehat{WA} = 64^\circ$$

Answer:

If \(m\widehat{VS} = 180^\circ\) and \(m\angle Z = 58^\circ\), find \(m\widehat{WA}\).

\(m\widehat{WA} =\) <blank>\(64^\circ\)</blank>