Question

using the figure shown to the right, find the value of each variable. lines that appear tangent are tangent.

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

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

Explanation:

Identify the angle relationships

The figure shows a circle with a circumscribed angle of \(50^\circ\) formed by two tangent lines.
The angle \(x^\circ\) is an inscribed angle subtending the same arc as the circumscribed angle, or we can use the relationship between the circumscribed angle and its intercepted arcs.
Let the major arc be \(A_{\text{major}}\) and the minor arc be \(A_{\text{minor}}\).
The circumscribed angle is related to the intercepted arcs by:

$$ 50^\circ = \frac{A_{\text{major}} - A_{\text{minor}}}{2} $$

Since the two lines are tangent, the sum of the major and minor arcs is \(360^\circ\):

$$ A_{\text{major}} + A_{\text{minor}} = 360^\circ $$

Solve for the intercepted arcs

We can solve the system of equations:

$$ A_{\text{major}} - A_{\text{minor}} = 100^\circ $$

$$ A_{\text{major}} + A_{\text{minor}} = 360^\circ $$

Adding the two equations:

$$ 2A_{\text{major}} = 460^\circ \implies A_{\text{major}} = 230^\circ $$

Subtracting the equations:

$$ 2A_{\text{minor}} = 260^\circ \implies A_{\text{minor}} = 130^\circ $$

Find the value of x

The angle \(x^\circ\) is a chord-tangent angle (or inscribed-like angle) that intercepts the minor arc \(A_{\text{minor}}\).
The measure of this angle is half the measure of its intercepted arc:

$$ x = \frac{A_{\text{minor}}}{2} = \frac{130}{2} = 65 $$

Find the value of y

The angle \(y^\circ\) is formed by a tangent and a secant (chord).
The intercepted arc on the far side is given as \(150^\circ\).
The angle \(y^\circ\) intercepts this arc of \(150^\circ\).
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc:

$$ y = \frac{150}{2} = 75 $$

Answer:

Using the figure shown to the right, find the value of each variable. Lines that appear tangent are tangent.

\(x =\) <blank>65</blank> (Simplify your answer. Do not include the degree symbol in your answer.)

\(y =\) <blank>75</blank> (Simplify your answer. Do not include the degree symbol in your answer.)