Yes, there are several theorems on angles formed by tangent and secant lines that can help find the measure of an arc. Here are two commonly used theorems:

1. The Angle-Arc Theorem: This theorem states that the measure of an angle formed by a tangent line and a chord is equal to half the measure of the intercepted arc. Mathematically, we can express this as:

  m∠A = 1/2mAB, where ∠A is the angle formed by tangent line AC and chord AB, and mAB is the measure of arc AB.

2. The Intersecting Secant Angles Theorem: This theorem states that the measure of an angle formed by two intersecting secant lines is equal to half the difference of the measures of the intercepted arcs. Mathematically, we can express this as:

  m∠A = 1/2(mCD - mBD), where ∠A is the angle formed by intersecting secant lines AC and BD, mCD is the measure of arc CD intercepted by AC, and mBD is the measure of arc BD intercepted by BD.

Using these theorems, we can find the measure of both the smaller and larger arcs. To find the measure of the smaller arc, we simply subtract the measure of the angle from the measure of the larger arc. For example, if we know that m∠A = 60 degrees and mBD = 80 degrees, we can use the Intersecting Secant Angles Theorem to find mCD, and then subtract 60 from mCD to find the measure of the smaller arc.

I hope that helps! Let me know if you have any further questions.

Answer:

Yes, there are theorems on angles formed by tangent and secant lines that can help you find the measure of an arc, both the smaller and larger ones. Here are two common theorems that can be used:

1. If a tangent line and a secant line intersect at a point on the circle, then the measure of the angle formed is equal to half the measure of the intercepted arc. This theorem is known as the "tangent-secant angle theorem".

To use this theorem to find the measure of an arc, you can follow these steps:

- Identify the tangent line and the secant line that intersect at a point on the circle.

- Determine which arc is intercepted by the secant line.

- Multiply the angle measure by 2 to find the measure of the intercepted arc.

2. If two secant lines intersect outside the circle, then the measure of the angle formed is equal to half the difference of the measures of the intercepted arcs. This theorem is known as the "secant-secant angle theorem".

To use this theorem to find the measure of an arc, you can follow these steps:

- Identify the two secant lines that intersect outside the circle.

- Determine which arcs are intercepted by the secant lines.

- Subtract the smaller arc measure from the larger arc measure.

- Divide the difference by 2 to find the measure of the angle formed.

Remember that the measure of a full circle is 360 degrees. If you have the measure of one arc, you can find the measure of the other arc by subtracting it from 360 degrees.

Step-by-step explanation: