27. Two tangents to a circle form an angle of 74°. What is the degree measure of the intercepted arc.

When two tangents are drawn to a circle from a common external point, they create an angle at that point, and this angle is supplementary to the angle subtended by the intercepted arc at the center of the circle.

Given:
- Angle between the two tangents = 74°

Let the measure of the intercepted arc be \( x \).

The relationship between the angle between the tangents and the measure of the intercepted arc is given by:
\[
x = 360^\circ - 2 \times \text{Angle between the tangents}
\]

Substituting the given angle:
\[
x = 360^\circ - 2 \times 74^\circ
\]
\[
x = 360^\circ - 148^\circ = 212^\circ
\]

So, the degree measure of the intercepted arc is **212°**.

Would you like any details or have any questions? 

Here are 5 related questions to expand on this topic:

1. How do you find the measure of an angle formed by two secants intersecting outside the circle?
2. What is the relationship between an inscribed angle and its intercepted arc?
3. How would the solution change if the angle between the tangents was given as 90°?
4. Can you explain the concept of a tangent to a circle and its properties?
5. How do you find the measure of an angle formed by a tangent and a chord?

**Tip:** Always remember that the angle formed between two tangents from a common point is supplementary to the angle subtended by the intercepted arc at the center of the circle.

Discover how to find the degree measure of the intercepted arc when two tangents form an angle of 74° with a detailed step-by-step solution. This problem explores circle geometry, tangents, and the angle sum property of circles. Suitable for high school students.

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