Circles and Angles (Outside, Inside, On)

Summary

This video explains the relationship between angles and arcs in circles, focusing on how to calculate angles formed by two arcs. The presenter demonstrates how to find the angle between two arcs by calculating half of the sum of the two arcs. The video also covers solving for an unknown arc and addresses both inscribed and central angles. A key takeaway is the formula: Angle = 1/2 (Arc 1 + Arc 2), applicable for different types of circle angles. The presenter provides clear examples and concludes by showing how the same formula applies to central angles with extended sides.

Takeaways

• 😀 The angle formed by two intersecting chords is half the sum of the arcs they subtend.
• 😀 To find the measure of an angle formed by two intersecting chords, use the formula: (Arc1 + Arc2) / 2.
• 😀 When two chords intersect inside a circle, the angles formed are determined by the arcs of the circle.
• 😀 The formula for the angle between two chords works for both intersecting chords inside the circle and those intersecting on the circle's circumference.
• 😀 The formula can be simplified by subtracting known arc measures from 360° to find the remaining arc's sum.
• 😀 If two arcs are known, the sum of the two arcs can be used to find the angle between the chords.
• 😀 In some problems, it's not the angle you're solving for, but the unknown arc, and the same formula applies.
• 😀 When an angle's measure involves an unknown arc, the formula can be adjusted by multiplying both sides by 2 for ease of calculation.
• 😀 Inscribed angles follow the same rule, where the angle is half the sum of the arcs it intercepts.
• 😀 Central angles also follow the same principle; however, to apply the formula, extend the sides of the central angle to form vertical angles.

Q & A

What is the formula for finding an angle in a circle when two arcs are involved?

The angle is equal to half the sum of the two arcs it intercepts. This means that the angle is calculated as (Arc 1 + Arc 2) / 2.

How can the formula be applied when one of the arcs is missing?

If one of the arcs is missing, you can subtract the sum of the known arcs from 360° to find the unknown arc, and then apply the formula to find the angle.

What is the relationship between a central angle and intercepted arcs?

A central angle's measure is equal to the measure of the intercepted arc. However, when the sides of the central angle are extended, the angle and its vertical opposite angle are still related to the arcs, following the same formula.

How do you handle the situation where the angle is already known, and you need to find one of the arcs?

To find the missing arc, multiply both sides of the equation by 2 to eliminate the fraction. Then, subtract the known arc from the total to find the missing one.

What is the importance of vertical angles when working with central angles?

Vertical angles help determine that if you extend the sides of the central angle, the opposite angle will have the same measure, allowing you to apply the same formula for central angles.

What is the method for finding the angle when the two arcs are given, but the angle is unknown?

To find the angle, sum the two arcs and divide by 2. This will give you the angle formed by those two arcs.

What happens if you subtract the sum of the arcs from 360°?

Subtracting the sum of the given arcs from 360° leaves you with the combined measure of the remaining arcs. This can then be used in the angle formula to determine the missing angle.

Why does multiplying both sides of the equation by 2 simplify the problem?

Multiplying by 2 removes the fraction (1/2) from the equation, making it easier to solve for the unknown value.

Are there any exceptions to the rule about angles and arcs in a circle?

No, the formula for calculating angles based on the sum of intercepted arcs applies to both inscribed and central angles, although the method for central angles requires extending the sides.

How does the instructor suggest handling problems with more than one unknown in arcs and angles?

The instructor suggests breaking down the problem by isolating the unknown and simplifying the equation. If necessary, subtract known values from 360° or use basic algebra to solve for the missing parts.