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.

We begin by recapping the definitions of the different types of lines that meet or intersect in a circle.

  • A chord of a circle is a line segment whose endpoints both lie on the circle’s circumference.
  • A secant is a line that intersects a circle at exactly two points. A secant can be thought of as a chord that has been extended indefinitely in both directions.
  • A tangent is a line that touches a circle at only one point.

These three types of lines are illustrated in the figure below.

The focus of this explainer is on determining the measures of the angles formed when two of these lines intersect, either inside or outside a circle. The measures of these angles are related to the measures of the arcs intercepted by the lines which form their sides. We should recall that the measure of an arc is defined to be equal to the measure of its central angle, as illustrated in the figure below.

We first consider intersections inside a circle. Our first definition concerns the measures of the angles formed by intersecting chords.

Theorem: Angles between Intersecting Chords

The measure of the angle formed by two chords that intersect inside a circle is equal to one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Consider the angles formed by the intersection of the chords 𝐴𝐡 and 𝐢𝐷 in the figure below.

The arc intercepted by angle π‘₯ is 𝐴𝐢. The arc intercepted by its vertically opposite angle is 𝐡𝐷. Hence, by the theorem of the angles between intersecting chords, π‘₯=12ο€»π‘šο€»π΄πΆο‡+π‘šο€Ίπ΅π·ο†ο‡.

For angle 𝑦, the arcs intercepted by this angle and its vertical angle are 𝐡𝐢 and 𝐴𝐷. Hence, 𝑦=12ο€»π‘šο€Ίπ΅πΆο†+π‘šο€»π΄π·ο‡ο‡.

The same result can also be applied to find the measure of the angle formed when two secants intersect inside a circle, or a secant and a chord. This is possible because a secant is an extension of a chord indefinitely in both directions.

In our first example, we will demonstrate how to apply this result to find the measure of the angle between two intersecting chords when given the measures of the two intercepted arcs.

Example 1: Finding the Measure of an Inscribed Angle between Two Intersecting Chords given the Inscribed Arcs

Find π‘₯.

Answer

From the figure, we see that the line segments 𝐴𝐡 and 𝐢𝐷 are each chords of the circle, as both endpoints of each line segment lie on the circle’s circumference. The value we are asked to calculate, π‘₯, is the measure of one of the angles formed at the point where these two chords intersect. We therefore recall the theorem of the angles between intersecting chords: β€œThe measure of the angle formed by two chords that intersect inside a circle is equal to one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.”

The intercepted arcs for angle π‘₯ and its vertical angle are 𝐴𝐢 and 𝐡𝐷. Hence, π‘₯=12ο€»π‘šο€»π΄πΆο‡+π‘šο€Ίπ΅π·ο†ο‡.

Substituting π‘šο€»π΄πΆο‡=73∘ and π‘šο€Ίπ΅π·ο†=133∘ and simplifying gives π‘₯=12(73+133)=12Γ—206=103.∘∘∘∘

We now consider the angles formed by intersections outside a circle. In this case, the two lines intersecting could be both tangents or both secants or one of each.

Theorem: Angles between Intersecting Secants and Tangents

The measure of the angle formed by two secants, two tangents, or a secant and a tangent that intersect at a point outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.

In the figure below, we illustrate this result for the angle formed by the intersection of two secants, ⃖⃗𝐴𝐢 and ⃖⃗𝐴𝐸.

The minor arc intercepted by the two secants is 𝐡𝐷 and the major arc is 𝐢𝐸. Hence, by the theorem of angles between intersecting secants, π‘₯=12ο€Ίπ‘šο€ΊπΆπΈο†βˆ’π‘šο€Ίπ΅π·ο†ο†.

In the same way, we illustrate the result for the intersection of two tangents, ⃖⃗𝐴𝐡 and ⃖⃗𝐴𝐢: π‘₯=12ο€Όπ‘šο€Όπ΅π·πΆοˆβˆ’π‘šο€Ίπ΅πΆο†οˆ.

Note that when two tangents intersect at a point outside a circle, the major and minor intercepted arcs together form the entire circumference of the circle. Hence, the measures of the two intercepted arcs sum to 360∘. This is important to remember as we may only be given the measure of one of the intercepted arcs and expected to calculate the other by applying this knowledge.

We now consider an example in which we will find the measure of the angle between two secants that intersect outside a circle given the measures of the two intercepted arcs.

Example 2: Finding the Measure of the Inscribed Angle between Two Secants given the Measure of the Two Intercepted Arcs

Find the value of π‘₯.

Answer

The line segments 𝐴𝐸 and 𝐢𝐸 are both segments of secants of the circle, as they each intersect the circle at exactly two points. The two secant segments intersect at a point outside the circle and the value we are asked to calculate is the measure of the angle formed. Hence, we recall the theorem of the angles between intersecting secants: β€œThe measure of the angle formed by two secants that intersect at a point outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.”

The two intercepted arcs are 𝐴𝐢 and 𝐡𝐷. As 𝐴𝐢 has the larger measure, the positive difference is found by subtracting the measure of 𝐡𝐷 from that of 𝐴𝐢. Hence, π‘₯=12ο€»π‘šο€»π΄πΆο‡βˆ’π‘šο€Ίπ΅π·ο†ο‡.∘

Substituting the measures of the two arcs as given in the figure and simplifying gives π‘₯=12(144βˆ’71)=12Γ—73=36.5.∘∘∘∘∘

The value of π‘₯ is 36.5.

Note that, in the previous problem, the value of π‘₯ was purely numerical: our answer was 36.5, not 36.5∘. We contrast this with example 1, in which our solution was π‘₯=103∘. This is due to the difference in whether the unit of measurement (degrees) was included when labeling the angle: in example 1, the angle was labeled simply as π‘₯, whereas in our second example, the angle was labeled as π‘₯∘.

We have now seen examples of how to calculate the measure of the angle between two chords and the measure of the angle between two secants, given the measures of the two intercepted arcs. It is also possible to work backward given the measure of the angle between two chords, secants, or tangents to determine the measure of one or both intercepted arcs, provided we are given sufficient other information. In more complex problems, this may also require us to form and solve an algebraic equation, as we will see in our next example.

Example 3: Finding the Measure of a Major Arc given the Measures of the Minor Arc and the Inscribed Angle between Two Tangents to Those Arcs

Given that π‘₯∘ is the measure of the major arc 𝐡𝐢, find the value of π‘₯.

Answer

Upon inspection of the figure, we observe that there are two tangents, ⃖⃗𝐴𝐡 and ⃖⃗𝐴𝐢, drawn from the same external point to a circle. We are asked to calculate the measure of the major arc intercepted by these two tangents. We recall the theorem of the angles between intersecting tangents: β€œThe measure of the angle formed by two tangents that intersect at a point outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.”

If we imagine a point 𝐷 on the circle’s circumference anywhere on the major arc connecting 𝐡 and 𝐢, we can express this result for this problem as π‘šβˆ (𝐢𝐴𝐡)=12ο€Όπ‘šο€Όπ΅π·πΆοˆβˆ’π‘šο€Ίπ΅πΆο†οˆ.

In the figure, we are given the measure of the angle between the two tangents and an algebraic expression for the measure of the major arc which we are now referring to as 𝐡𝐷𝐢. To find an expression for the measure of the minor arc, we recall that the measure of the total circumference of a circle is 360∘. Hence, the measure of the minor arc, 𝐡𝐢, is equal to (360βˆ’π‘₯)∘.

We can now form an equation in π‘₯ by substituting these values and expressions into the formula above. The unit of measurement is the same for each expression and so can be omitted. Substituting π‘₯ for the measure of the major arc, (360βˆ’π‘₯) for the measure of the minor arc, and 64 for the measure of the angle between the two tangents gives 12(π‘₯βˆ’(360βˆ’π‘₯))=64.

To solve for π‘₯, we first multiply both sides of the equation by 2 and then distribute the parentheses: (π‘₯βˆ’(360βˆ’π‘₯))=1282π‘₯βˆ’360=128.

Finally, we add 360 to each side of the equation and then divide both sides by 2: 2π‘₯=488π‘₯=244.

Let us now consider another example in which we are required to form and solve an algebraic equation by relating the measure of the angle between a secant and a tangent to the measures of the two intercepted arcs. Both of these arc measures will be given as linear expressions of the unknown we are required to determine.

Example 4: Finding the Measure of the Two Arcs Inscribed between Secants given the Inscribed Angle

Given that, in the shown figure, 𝑦=(π‘₯βˆ’2) and 𝑧=(2π‘₯+2), determine the value of π‘₯.

Answer

From the figure, we see that the line segment 𝐴𝐡 is a tangent to the circle as it intersects the circle at only one point. The line segment 𝐴𝐷 is a secant segment as it intersects the circle at exactly two points and its endpoint is on the circumference of the circle. These two line segments intersect at a point outside the circle and we are given the measure of the angle formed by their intersection. We recall the theorem of angles between intersecting secants and tangents: β€œThe measure of the angle formed by a secant and a tangent that intersect at a point outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.”

From the figure, we observe that the major intercepted arc is 𝐡𝐷, and the minor intercepted arc is 𝐡𝐢. Hence, we can form an equation using the measures of these two arcs and the measure of the angle of intersection of the secant and tangent: 50=12(π‘§βˆ’π‘¦).

We are given expressions for both 𝑦 and 𝑧 in terms of a third variable, π‘₯, the value of which we are required to calculate. Substituting 𝑧=2π‘₯+2 and 𝑦=π‘₯βˆ’2 into the equation above gives an equation in π‘₯ only: 50=12((2π‘₯+2)βˆ’(π‘₯βˆ’2)).

We now solve this equation to find π‘₯. Although not entirely necessary, we will begin by swapping the two sides so that the unknown appears on the left-hand side. We then simplify within the parentheses to give the following: 12(2π‘₯+2βˆ’π‘₯+2)=5012(π‘₯+4)=50.

Multiplying both sides of the equation by 2 gives π‘₯+4=100.

Finally, subtracting 4 from each side of the equation gives π‘₯=96.

We have now considered four examples in which we have demonstrated the application of the two key theorems to both numerical and algebraic problems. The results we have introduced in this explainer can also be applied to more complex problems involving other geometric shapes inscribed in circles. Let us now consider an example in which a regular pentagon is inscribed in a circle and we are required to find the measure of the angle between two tangents to the circle.

Example 5: Finding the Angle between Two Tangents by Using the Properties of Tangents to a Circle and Regular Polygons

𝐴𝐡𝐢𝐷𝐸 is a regular pentagon drawn inside the circle 𝑀, ⃖⃗𝐴𝑋 is a tangent to the circle at 𝐴, and ⃖⃗𝐸𝑋 is a tangent to the circle at 𝐸. Find π‘šβˆ π΄π‘‹πΈ.

Answer

Upon inspection of the diagram, we see that the angle 𝐴𝑋𝐸 is the angle formed by the intersection of the two tangents ⃖⃗𝐴𝑋 and ⃖⃗𝑋𝐸. We therefore recall the theorem of the angles between intersecting tangents: β€œThe measure of the angle formed by two tangents that intersect at a point outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.”

We may find it helpful to add color to the diagram to help identify the intercepted arcs, as shown below.

We will refer to the major arc, shown in pink, as 𝐴𝐡𝐸, and the minor arc, shown in orange, as 𝐴𝐸. Hence the measure of angle 𝐴𝑋𝐸 is given by π‘šβˆ π΄π‘‹πΈ=12ο€»π‘šο€»π΄π΅πΈο‡βˆ’π‘šο€Ίπ΄πΈο†ο‡.

We have not been given the measures of any angles or any arcs in the figure. Instead, we recall that the pentagon 𝐴𝐡𝐢𝐷𝐸 is regular. It can, therefore, be divided into five congruent triangles by drawing in the radii from each vertex of the pentagon to the center of the circle. We illustrate one such triangle by drawing in the radii 𝐴𝑀 and 𝐸𝑀 in the figure below.

The measure of the major arc 𝐴𝐡𝐸 is equal to the reflex angle at the center of the circle. The measure of the minor arc 𝐴𝐸 is equal to the acute angle at the same point. We recall that angles around a point sum to 360∘. As the pentagon is regular and the five triangles are congruent, the measure of the acute angle 𝐸𝑀𝐴 can be found by dividing 360∘ by 5: π‘šβˆ πΈπ‘€π΄=3605=72.∘∘

Hence, the measure of the minor arc 𝐴𝐸 is 72∘. The measure of the major arc can be found by subtracting this value from 360∘ to give 288∘.

Substituting the measures of the two arcs into the formula above, we have π‘šβˆ π΄π‘‹πΈ=12(288βˆ’72)=12Γ—216=108.∘∘∘∘

In more complex problems with multiple intersecting line segments, we may need to apply more than one of the theorems introduced in this explainer. We may also need to use results relating to other types of angles in circles. An inscribed angle has its vertex on the circumference of a circle and sides that contain chords of the circle. We define below the relationship between the measure of an inscribed angle and its intercepted arc.

Definition: Measure of an Inscribed Angle

The measure of an angle inscribed in a circle is equal to one-half the measure of its intercepted arc.

For the figure below, this result can be expressed as π‘šβˆ π΄πΆπ΅=12π‘šο€Ίπ΄π΅ο†.

We now consider one final example: a multistep problem in which we apply both the theorem of the angles between intersecting chords and the theorem of the angles between intersecting secants, in addition to our knowledge of inscribed angles.

Example 6: Finding the Measure of an Angle given the Measures of Its Major and Minor Arcs

Find π‘₯.

Answer

Upon inspection of the figure, we see that π‘₯∘ is the measure of the angle formed by the intersection of the two chords 𝐡𝐸 and 𝐢𝐷 inside a circle. Hence, by the theorem of the angles between intersecting chords, the measure of this angle is equal to half the sum of the intercepted arcs: π‘₯=12ο€Ίπ‘šο€ΊπΆπΈο†+π‘šο€Ίπ΅π·ο†ο†.∘

Next, we note that one of the angles whose measures we are given, angle 𝐡𝐴𝐷, is the angle formed by the intersection of the secant segments 𝐴𝐢 and 𝐴𝐸 outside the circle. Hence, recalling that the measure of such an angle is equal to half the positive difference of the intercepted arcs, we have 40=12ο€Ίπ‘šο€ΊπΆπΈο†βˆ’π‘šο€Ίπ΅π·ο†ο†.∘

We now have two linear simultaneous equations involving the measures of 𝐢𝐸 and 𝐡𝐷, but we currently have insufficient information to solve them. The other information given in the diagram is the measure of the inscribed angle 𝐡𝐸𝐷. Recalling that the measure of an inscribed angle is equal to half the measure of its intercepted arc, we can calculate the measure of 𝐡𝐷: 12π‘šο€Ίπ΅π·ο†=30π‘šο€Ίπ΅π·ο†=2Γ—30=60.∘∘∘

We can now substitute this value into our second equation, which will enable us to find the measure of 𝐢𝐸. We will then be able to substitute the measures of both arcs into our first equation to determine π‘₯.

Substituting π‘šο€Ίπ΅π·ο†=60∘ into our second equation gives 12ο€Ίπ‘šο€ΊπΆπΈο†βˆ’60=40.∘∘

We solve to determine π‘šο€ΊπΆπΈο† by first multiplying each side of the equation by 2 and then adding 60∘ to each side: π‘šο€ΊπΆπΈο†βˆ’60=80π‘šο€ΊπΆπΈο†=140.∘∘∘

Finally, we find π‘₯ by calculating half the sum of the measures of 𝐢𝐸 and 𝐡𝐷: π‘₯=12ο€Ίπ‘šο€ΊπΆπΈο†+π‘šο€Ίπ΅π·ο†ο†=12(140+60)=12Γ—200=100.∘∘∘∘∘

Hence, π‘₯=100.

Let us finish by recapping some key points from this explainer.

Key Points

  • The measure of the angle formed by two chords that intersect inside a circle is equal to one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
  • The measure of the angle formed by two secants, two tangents, or a secant and a tangent that intersect at a point outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.
  • The measure of an angle inscribed in a circle is equal to one-half the measure of its intercepted arc.
  • These results can be applied to both numerical and algebraic problems to calculate the measures of angles resulting from the intersection of two chords, two secants, two tangents, or a tangent and a secant in a circle.

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