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The given problem involves two intersecting secants and a circle, where the objective is to find the value of \( x \) (an angle) in degrees.

To solve this problem, we can use the **Secant Angle Theorem**, which states that the measure of an angle formed by two intersecting secants outside a circle is equal to **half the difference of the intercepted arcs**.

### Formula:
\[
\theta = \frac{1}{2} \left( \text{larger intercepted arc} - \text{smaller intercepted arc} \right)
\]

In this case:
- The larger intercepted arc is \( 74.2^\circ \)
- The smaller intercepted arc is \( 30.3^\circ \)
- The angle \( x^\circ \) is the one formed outside the circle.

### Calculation:
\[
x = \frac{1}{2} \left( 74.2^\circ - 30.3^\circ \right)
\]

Let's calculate that.The value of \( x \) is approximately \( 22.0^\circ \), rounded to one decimal place.

Would you like further details on this solution or have any other questions?

Here are 5 related questions for practice:
1. How would the formula change if the secants intersect inside the circle instead of outside?
2. What is the significance of the Secant Angle Theorem in geometry?
3. How do you find the measure of an angle formed by a tangent and a secant?
4. Can you derive the formula used in this problem from basic circle theorems?
5. How would the calculation change if one of the secants became a tangent?

**Tip:** When working with angles in circles, always check whether secants, tangents, or chords are involved to apply the correct geometric theorem.

The architect designed a building front with an arching roof based off a circle and two intersecting secants. The values of the intercepted arcs are 74.2° and 30.3°. Find the angle formed outside the circle, x.

Learn how to find the exterior angle formed by two intersecting secants outside a circle using the Secant Angle Theorem. The intercepted arcs are 74.2° and 30.3°, and the angle is calculated using the geometric formula. Suitable for high school students.

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