Question

using the figure shown to the right, find the value of each variable.

\\(x =\\) (simplify your answer. do not include the degree symbol in your answer.)

Explanation:

Identify the given values and geometric relationships

We are given a circle with intersecting chords.

• One inscribed angle is \(58^\circ\), which intercepts an arc.
• Another inscribed angle is \(50^\circ\), which intercepts another arc.
• An arc at the bottom is labeled with a measure of \(60^\circ\).
• We need to find the value of the variable \(x\), which represents an angle formed by two intersecting chords inside the circle.

Apply the Inscribed Angle Theorem

Using the Inscribed Angle Theorem:

• The inscribed angle of \(58^\circ\) intercepts the far-right arc. Therefore, the measure of this intercepted arc is:

$$ 2 \times 58^\circ = 116^\circ $$

• The inscribed angle of \(50^\circ\) intercepts the far-left arc. Therefore, the measure of this intercepted arc is:

$$ 2 \times 50^\circ = 100^\circ $$

Determine the measure of the top arc

The total measure of a circle's circumference is \(360^\circ\).

• The four arcs making up the circle are the top arc, the right arc (\(116^\circ\)), the bottom arc (\(60^\circ\)), and the left arc (\(100^\circ\)).
• Let the measure of the top arc be \(T\):

$$ T = 360^\circ - (116^\circ + 60^\circ + 100^\circ) = 360^\circ - 276^\circ = 84^\circ $$

Apply the Angles of Intersecting Chords Theorem

The angle \(x\) is an interior angle formed by two intersecting chords.

• The measure of an angle formed by two intersecting chords inside a circle is half the sum of the measures of their intercepted arcs.
• The arcs intercepted by the angle \(x\) and its vertical angle are the top arc (\(84^\circ\)) and the bottom arc (\(60^\circ\)).
• Therefore, we calculate \(x\) as:

$$ x = \frac{84 + 60}{2} = \frac{144}{2} = 72 $$

Answer:

Using the figure shown to the right, find the value of each variable.

\(x =\) <blank>72</blank>