In this explainer, we will learn how to find the measures of angles resulting from the intersection of two chords, two secants, two tangents, or tangents and secants in a circle.

Consider two chords and in a circle with center that intersect inside the circle at . What can we say about the measure of the angle between the two chords, ?

We are going to prove a relationship between and the measures of and . For this, we first draw the segment between points and .

As points , , and are on a line, and are supplementary, which means that their measures add up to . is itself supplementary to the sum of and since these three angles are the angles of triangle . Therefore, with , , and , we have

The measures of inscribed angles and , and , are half the measures of the arcs they are subtended by, and respectively. Hence, we have

In the same way, we can prove that

### Property: Measure of the Angle between Intersecting Chords in a Circle

If two chords intersect at a point inside a circle, then the measure of the angle between the two chords equals half of the sum of the measures of the two arcs opposite the angle.

Let us look at how to apply this property of intersecting chords inside a circle with our first example.

### Example 1: Finding the Angle of Intersecting Chords inside a Circle

Find .

### Answer

We have two chords, and , that intersect inside the circle. We are asked to find , which is the measure of the angle between the chords.

Remember that if two chords intersect at a point inside a circle, then the measure of the angle between the two chords equals half of the sum of the measures of the two arcs opposite the angle.

The two arcs opposite the angle of measure are and . Their measures are and respectively.

Hence, we have

Let us now consider two secants (remember, a secant is the line extension of a chord) of a circle that intersect outside the circle and see what we find about the measure of the angle between the two secants.

In the following figure, secants and intersect outside the circle at point .

Angle is supplementary to angle , which is itself supplementary to since , , and are the 3 angles of triangle . Therefore, we have

Rearranging to make the subject gives

The measures of the inscribed angles and are half the measures of the arcs they are subtended by, and respectively. Hence, we have

### Property: Measure of the Angle between Intersecting Secants outside a Circle

If two secants intersect at a point outside a circle, then the measure of the angle between the two secants equals half of the positive difference between the measures of both arcs intercepted by the sides of the angle.

Let us see how to use this property with the following example.

### Example 2: Finding the Angle of Intersecting Secants

Find .

### Answer

We have two secants (remember, a secant is the line extensions of a chord) of a circle, and , that intersect outside the circle at point . We need to find the value of , which is the measure of angle between the two secants.

Remember that if two secants intersect at a point outside a circle, then the measure of the angle between the two secants equals half of the positive difference between the measures of both arcs intercepted by the sides of the angle.

Hence, we have

It is worth noting that we need to subtract the measure of the minor arc from that of the measure of the major arc since the difference needs to be positive as angle measures are positive.

Now, consider two tangents at points and of a circle with center that intersect at point outside the circle.

As the sum of the angles in quadrilateral is , we have

Furthermore, radius is perpendicular to the tangent to the circle at , so . Similarly, radius is perpendicular to the tangent to the circle at , so :

Angle is the central angle subtended by . Their measures are therefore equal, and we have

From this relationship, we can find a relationship involving both the minor and major arcs . Let us call these arcs and , respectively, as shown in the following diagram.

Multiplying both sides of our formula by 2 gives

Since the two intersecting tangents split the whole circle into two arcs, we have which gives

Substituting in for the first into our previous equation, we find that

Dividing both sides by 2 gives

### Property: Measure of the Angle between Intersecting Tangents outside a Circle

If two tangents intersect at a point outside a circle, then the measure of the angle between the two tangents is the measure of the minor arc between the two points of contact with the tangents subtracted from .

The measure of the angle between the two tangents is also half the difference between the major and minor arcs between the two points of contact with the tangents.

Let us now look at an example where we need to apply this property.

### Example 3: Finding the Angle of Intersecting Tangents

Find .

### Answer

The rays and are tangents to the circle at and respectively. We need to find the measure of the angle between the two tangents.

Remember that if two tangents intersect at a point outside a circle, then the measure of the angle between the two tangents is given either by the measure of the minor arc between the two points of contact with the tangents subtracted from or by half the difference between the major and minor arcs between the two points of contact with the tangents.

The measure of the minor arc is given; it is .

Hence, we have or as the major arc is , we find that

Let us use look at another example with two intersecting tangents.

### Example 4: Solving a Multistep Problem Involving Two Intersecting Tangents

In the given figure, find the values of and .

### Answer

Rays and are tangents to the circle at and respectively. We need to find the values of and knowing that is the measure of the angle between the two tangents and the measures of the arcs intercepted by the two sides of the angle are and .

Remember that if two tangents intersect at a point outside a circle, then the measure of the angle between the two tangents equals the measure of the minor arc between the two points of contact with the tangents subtracted from .

Therefore, we can write that

As the sum of the minor and major arcs is the whole circle, we have

We have found that and .

Let us finally consider the case of a tangent and a secant intersecting outside a circle at point .

Angle is supplementary to angle , which is itself supplementary to since , , and are the three angles of triangle . Therefore, we have

Rearranging gives

To find , we write that the sum of the angles in quadrilateral is .

Noting that is a right angle since is a radius and is the tangent at , and with the angle measures indicated in the diagram above, we have

The measure of central angle , , is the same as that of .

In addition, considering the sum of angles in triangle , we find that

Finally, triangle is isosceles (two sides are radii of the circle); therefore,

Therefore, substituting , , and into equation (2), we find that

After simplifying, we find that

Now, we can come back to equation (1), and note that the measure of inscribed angle is half that of the it is subtended by.

Hence, we find that

### Property: Measure of the Angle between a Tangent and a Secant Intersecting outside a Circle

If a tangent and a secant intersect at a point outside a circle, then the measure of the angle between them equals half of the difference between the measures of both arcs intercepted by the sides of the angle.

Let us see how to apply this property with our next example.

### Example 5: Finding the Angle of Intersection of a Tangent and a Secant with Linear Equations

Given that, in the shown figure, and , determine the value of .

### Answer

Segments and are, respectively, a secant of the circle at and and a tangent of the circle at , intersecting at and forming a angle. We need to determine the value of while we are given the measure of the arc intercepted by the secant and the tangent, and , in terms of .

Recall that if a tangent and a secant intersect at a point outside a circle, then the measure of the angle between them equals half of the difference between the measures of both arcs intercepted by the sides of the angle.

We can therefore write that

We can now substitute the expressions for and in terms of in the above equation, which gives

Let us now summarize what we have learned in this explainer.

### Key Points

- If two chords intersect at a point inside a circle, then the measure of the angle between the two chords equals half of the sum of the measures of the two arcs opposite the angle.
- If two secants, two tangents, or a secant and a tangent intersect at a point outside a circle, then the measure of the angle between them equals half of the positive difference between the measures of both arcs intercepted by the sides of the angle.
- The measure of the angle between two tangents is the measure of the minor arc between the two points of contact with the tangents subtracted from .
- The measure of the angle between the two tangents is also half the difference between the major and minor arcs between the two points of contact with the tangents.