Question

if \\(m\widehat{em} = 122^{\circ}\\) and \\(m\angle ucr = 130^{\circ}\\), find \\(m\widehat{ur}\\).

Explanation:

Identify the geometric relationship

The chords \(EU\) and \(MR\) intersect at point \(C\) inside the circle. The angle \(\angle UCR\) is formed by these intersecting chords.

Apply the intersecting chords theorem

The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs.

$$ m\angle UCR = \frac{m\overparen{UR} + m\overparen{EM}}{2} $$

Substitute the given values

We are given \(m\overparen{EM} = 122^\circ\) and \(m\angle UCR = 130^\circ\). Let \(x = m\overparen{UR}\).
Using Linear Equations:

$$ 130^\circ = \frac{x + 122^\circ}{2} $$

Solve for the unknown arc measure

Multiply both sides by 2:

$$ 260^\circ = x + 122^\circ $$

Subtract \(122^\circ\) from both sides:

$$ x = 260^\circ - 122^\circ = 138^\circ $$

Answer:

If \(m\overparen{EM} = 122^\circ\) and \(m\angle UCR = 130^\circ\), find \(m\overparen{UR}\).
\(m\overparen{UR} =\) <blank>\(138^\circ\)</blank>