Lesson Explainer: Angles of Intersecting Lines in a Circle Mathematics

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

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

As points 𝐴, 𝐸, and 𝐵 are on a line, 𝐵𝐸𝐶 and 𝐴𝐸𝐶 are supplementary, which means that their measures add up to 180. 𝐴𝐸𝐶 is itself supplementary to the sum of 𝐸𝐴𝐶 and 𝐴𝐶𝐸 since these three angles are the angles of triangle 𝐴𝐸𝐶. Therefore, with 𝑒=𝑚𝐵𝐸𝐶, 𝑎=𝑚𝐸𝐴𝐶, and 𝑐=𝑚𝐴𝐶𝐸, we have 𝑒=𝑎+𝑐.

The measures of inscribed angles 𝐸𝐴𝐶 and 𝐸𝐶𝐴, 𝑎 and 𝑐, are half the measures of the arcs they are subtended by, 𝐵𝐶 and 𝐴𝐷 respectively. Hence, we have 𝑒=12𝑚𝐵𝐶+12𝑚𝐴𝐷𝑒=12𝑚𝐵𝐶+𝑚𝐴𝐷.

In the same way, we can prove that 𝑚𝐷𝐸𝐵=12𝑚𝐵𝐷+𝑚𝐴𝐶.

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 𝑥.

Answer

We have two chords, 𝐴𝐵 and 𝐶𝐷, that intersect inside the circle. We are asked to find 𝑥, 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 𝑥 are 𝐴𝐶 and 𝐵𝐷. Their measures are 73 and 133 respectively.

Hence, we have 𝑥=12(73+133)𝑥=12(206)𝑥=103.

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 𝐴𝐵 and 𝐶𝐷 intersect outside the circle at point 𝐸.

Angle 𝐶𝐷𝐴 is supplementary to angle 𝐴𝐷𝐸, which is itself supplementary to 𝐴+𝐸 since 𝐴, 𝐸, and 𝐴𝐷𝐸 are the 3 angles of triangle 𝐴𝐷𝐸. Therefore, we have 𝑚𝐶𝐷𝐴=𝑚𝐴+𝑚𝐸.

Rearranging to make 𝑚𝐸 the subject gives 𝑚𝐸=𝑚𝐶𝐷𝐴𝑚𝐴.

The measures of the inscribed angles 𝐶𝐷𝐴 and 𝐴 are half the measures of the arcs they are subtended by, 𝐶𝐴 and 𝐵𝐷 respectively. Hence, we have 𝑚𝐸=12𝑚𝐶𝐴12𝑚𝐵𝐷𝑚𝐸=12𝑚𝐶𝐴𝑚𝐵𝐷.

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 𝑥.

Answer

We have two secants (remember, a secant is the line extensions of a chord) of a circle, 𝐴𝐶 and 𝐴𝐸, that intersect outside the circle at point 𝐴. We need to find the value of 𝑥, which is the measure of angle 𝐶𝐴𝐸 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 𝑚𝐶𝐴𝐸=12𝑚𝐶𝐸𝑚𝐵𝐷𝑥=12(13236)𝑥=12(96)𝑥=48.

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 𝐴 and 𝐵 of a circle with center 𝑀 that intersect at point 𝐶 outside the circle.

As the sum of the angles in quadrilateral 𝐴𝐶𝐵𝑀 is 360, we have 𝑚𝐴+𝑚𝐶+𝑚𝐵+𝑚𝑀=360.

Furthermore, radius 𝑀𝐴 is perpendicular to the tangent to the circle at 𝐴, so 𝑚𝐴=90. Similarly, radius 𝑀𝐵 is perpendicular to the tangent to the circle at 𝐵, so 𝑚𝐵=90: 90+𝑚𝐶+90+𝑚𝑀=360𝑚𝐶+𝑚𝑀=180𝑚𝐶=180𝑚𝑀.

Angle 𝑀 is the central angle subtended by 𝐴𝐵. Their measures are therefore equal, and we have 𝑚𝐶=180𝑚𝐴𝐵.

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

Multiplying both sides of our formula 𝑚𝐶=180𝑥 by 2 gives 2𝑚𝐶=3602𝑥=360𝑥𝑥.

Since the two intersecting tangents split the whole circle into two arcs, we have 𝑥+𝑦=360, which gives 𝑥=360𝑦.

Substituting in for the first 𝑥 into our previous equation, we find that 2𝑚𝐶=𝑦𝑥.

Dividing both sides by 2 gives 𝑚𝐶=12(𝑦𝑥).

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 180.

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 𝑥.

Answer

The rays 𝐴𝐶 and 𝐴𝐵 are tangents to the circle at 𝐶 and 𝐵 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 180 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 𝐵𝐶 is given; it is 151.

Hence, we have 𝑥=180151𝑥=29, or as the major arc is (360151)=209, we find that 𝑥=12(209151)=29.

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 𝑥 and 𝑦.

Answer

Rays 𝐴𝐶 and 𝐴𝐵 are tangents to the circle at 𝐶 and 𝐵 respectively. We need to find the values of 𝑥 and 𝑦 knowing that 𝑥 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 2𝑥 and 𝑦.

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 180.

Therefore, we can write that 𝑚𝐴=180𝑚𝐶𝐵𝑥=1802𝑥3𝑥=180𝑥=60.

As the sum of the minor and major arcs 𝐶𝐵 is the whole circle, we have 2𝑥+𝑦=3602×60+𝑦=360𝑦=360120𝑦=240.

We have found that 𝑥=60 and 𝑦=240.

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

Angle 𝐴𝐵𝐶 is supplementary to angle 𝐴𝐵𝐷, which is itself supplementary to 𝐷𝐴𝐵+𝐴𝐷𝐵 since 𝐷𝐴𝐵, 𝐴𝐷𝐵, and 𝐴𝐵𝐷 are the three angles of triangle 𝐴𝐵𝐷. Therefore, we have 𝑚𝐴𝐵𝐶=𝑚𝐷𝐴𝐵+𝑚𝐴𝐷𝐵.

Rearranging gives

𝑚𝐴𝐷𝐵=𝑚𝐴𝐵𝐶𝑚𝐷𝐴𝐵.(1)

To find 𝑚𝐷𝐴𝐵, we write that the sum of the angles in quadrilateral 𝑀𝐴𝐷𝐵 is 360.

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

𝑚+90+𝑑+𝑏+𝑏=360.(2)

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

In addition, considering the sum of angles in triangle 𝐴𝐵𝐷, we find that 𝑏=180(𝑎+𝑑).

Finally, triangle 𝐴𝑀𝐵 is isosceles (two sides are radii of the circle); therefore, 𝑏=𝑚𝑀𝐴𝐵=90𝑎.

Therefore, substituting 𝑚=𝑚𝐴𝐵, 𝑏=180(𝑎+𝑑), and 𝑏=90𝑎 into equation (2), we find that 𝑚𝐴𝐵+90+𝑑+180(𝑎+𝑑)+90𝑎=360.

After simplifying, we find that 𝑚𝐴𝐵2𝑎=0𝑎=12𝑚𝐴𝐵.

Now, we can come back to equation (1), 𝑚𝐴𝐵𝐶=𝑚𝐷𝐴𝐵+𝑚𝐴𝐷𝐵,𝑚𝐴𝐵𝐶=𝑎+𝑑, and note that the measure of inscribed angle 𝐴𝐵𝐶 is half that of the 𝐴𝐶 it is subtended by.

Hence, we find that 𝑑=𝑚𝐴𝐵𝐶𝑎𝑑=12𝑚𝐴𝐶12𝑚𝐴𝐵𝑑=12𝑚𝐴𝐶𝑚𝐴𝐵.

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, 𝑦=(𝑥2) and 𝑧=(2𝑥+2), determine the value of 𝑥.

Answer

Segments 𝐴𝐷 and 𝐴𝐵 are, respectively, a secant of the circle at 𝐶 and 𝐷 and a tangent of the circle at 𝐵, intersecting at 𝐴 and forming a 50 angle. We need to determine the value of 𝑥 while we are given the measure of the arc intercepted by the secant and the tangent, 𝐶𝐵 and 𝐵𝐷, in terms of 𝑥.

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 𝑚𝐷𝐴𝐵=12𝑚𝐷𝐵𝑚𝐶𝐵50=12(𝑧𝑦).

We can now substitute the expressions for 𝑧 and 𝑦 in terms of 𝑥 in the above equation, which gives 50=12((2𝑥+2)(𝑥2))50=12(2𝑥+2𝑥+2)50=12(𝑥+4)100=𝑥+4𝑥=96.

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 180.
  • 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.

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