When two chords of a circle are parallel are the arcs they congruent How about the arcs they cut off explain?

In this explainer, we will learn how to use the parallel chords and the parallel tangents and chords of a circle to deduce the equal measures of the arcs between them and find missing lengths or angles.

We begin by recapping some of the key terminology for circles. Consider the following circle, centered at 𝑀.

A chord is a line segment whose endpoints lie on the circumference of the circle. In the diagram, 𝐴𝐵 is a chord.

Similarly, a tangent to a circle is a line that intersects the circle exactly once. In the diagram, ⃖⃗𝐶𝐷 is a tangent to the circle at point 𝑃.

When the lines are added to a circle, the points where they meet the circle partition the circumference into a number of arcs. For instance, there are two arcs between points 𝐴 and 𝐵. The shorter arc is known as the minor arc 𝐴𝐵 (this is the arc whose measure is less than 180∘), and the longer arc (the arc with a measure greater than 180∘) is called the major arc. We denote the minor arc from 𝐴 to 𝐵 more succinctly as 𝐴𝐵.

With these definitions in mind, we will now define and prove a theorem that links parallel chords and arcs in a circle.

The measures of the arcs between parallel chords of a circle are equal.

In the diagram, 𝐴𝐵 is parallel to 𝐶𝐷, so 𝑚𝐴𝐶=𝑚𝐵𝐷.

While it is outside the scope of this explainer to prove this theorem, it can be proven in a minimal number of steps using the inscribed angle conjecture and properties of angles in parallel lines. We will now apply this theorem alongside other properties of chords to find the measure of an arc.

In the given figure, if the measure of arc 𝐵𝐷=65∘, find the measure of arc 𝐶𝐷.

Answer

Recall that arcs formed by a pair of parallel chords are congruent. In the diagram, 𝐴𝐵 and 𝐶𝐷 are parallel chords, so the arcs formed are congruent. That is, 𝑚𝐴𝐶=𝑚𝐵𝐷=65∘.

Next, since 𝐴𝐵 is a chord that passes through the center of the circle, it is a diameter. Hence, the measure of arc 𝐴𝐵 is 180∘.

By splitting 𝐴𝐵 into three separate arcs, we can use this information to form and solve an equation for the measure of arc 𝐶𝐷: 𝑚𝐴𝐵=𝑚𝐴𝐶+𝑚𝐶𝐷+𝑚𝐵𝐷180=65+𝑚𝐶𝐷+65180=130+𝑚𝐶𝐷.∘∘∘∘∘

Hence, 𝑚𝐶𝐷=180−130=50.∘∘∘

In our next example, we will use this theorem alongside properties of angles to find a missing arc measure.

Find 𝑚𝐵𝐸 where 𝑀 is the center of the circle.

Answer

Recall that arcs formed by a pair of parallel chords are congruent. Since 𝐴𝐵 and 𝐶𝐸 are parallel, it follows that 𝑚𝐵𝐸=𝑚𝐴𝐶.

We can see that there is a pair of vertically opposite angles, ∠𝐷𝑀𝐵 and ∠𝐴𝑀𝐶. Since vertically opposite angles are equal, 𝑚∠𝐴𝑀𝐶=𝑚∠𝐷𝑀𝐵=39.∘

Then, since ∠𝐴𝑀𝐶 is a central angle subtended by 𝐴𝐶, 𝑚𝐴𝐶=𝑚∠𝐴𝑀𝐶=39.∘

We know that this is equal to 𝑚𝐵𝐸; hence, 𝑚𝐵𝐸=𝑚𝐴𝐶=39.∘

A useful corollary to the above theorem is that the reverse statement also holds. If the measures of the two arcs between two distinct chords are equal, then the chords are parallel.

There is one further property that holds for chords of equal lengths. That is, if two chords are equal in length, then the arcs between the endpoints of the chords will be equal in measure.

In the diagram, since 𝐴𝐵 and 𝐶𝐷 are equal in length, 𝑚𝐴𝐵=𝑚𝐶𝐷.

In our next example, we will demonstrate how to apply these properties.

In the given figure, the measure of 𝐴𝐵=62∘, the measure of 𝐵𝐶=110∘, and the measure of 𝐴𝐷=126∘. What can we conclude about 𝐴𝐷 and 𝐵𝐶?

1. They are parallel.
2. They are neither parallel nor perpendicular.
3. They are perpendicular.
4. They are the same length.
5. They are parallel and of the same length.

Answer

Let’s begin by adding the measure of each arc to the diagram.

Since the measure of each arc is the angle that the arc makes at the center of the circle, the sum of all arc measures is 360∘.

Hence, 𝑚𝐴𝐵+𝑚𝐵𝐶+𝑚𝐶𝐷+𝑚𝐴𝐷=36062+110+𝑚𝐶𝐷+126=360298+𝑚𝐶𝐷=360.∘∘∘∘∘∘∘

So, 𝑚𝐶𝐷=360−298=62.∘∘∘

Next, we recall that if the measures of the two arcs between two chords are equal, then the chords must be parallel. Since 𝑚𝐶𝐷=𝑚𝐴𝐵, then chords 𝐴𝐷 and 𝐵𝐶 are parallel, and the answer is A or E. We observe that since 𝑚𝐵𝐶≠𝑚𝐴𝐷, then these arcs are not congruent and hence the chords cannot be of equal length. The answer is option A.

We will now extend our idea of parallel chords to include a parallel chord and a tangent with the following theorem.

The measures of the arcs between a parallel chord and a tangent of a circle are equal.

In the diagram, 𝐴𝐵 is parallel to the tangent at 𝐶, hence 𝑚𝐴𝐶=𝑚𝐵𝐶.

Once again, while it is outside the scope of this explainer to prove this theorem, it can be proven in just a few steps using the alternate segment theorem. With the theorem stated, let’s demonstrate its application.

𝑀 is a circle, where 𝐴𝐵 is a chord and ⃖⃗𝐶𝐷 is a tangent. If 𝐴𝐵⫽⃖⃗𝐶𝐷 and the measure of 𝐴𝐵=72∘, find the measure of 𝐵𝐶.

Answer

Since 𝐴𝐵⫽⃖⃗𝐶𝐷, we will use the following theorem: the measures of the arcs between a parallel chord and tangent of a circle are equal. This means that 𝑚𝐴𝐶=𝑚𝐵𝐶. We are given that 𝑚𝐴𝐵=72∘, and we know that the sum of the measures of all the arcs that make up the circle is 360∘. Hence, 𝑚𝐴𝐶+𝑚𝐵𝐶+𝑚𝐴𝐵=360𝑚𝐴𝐶+𝑚𝐵𝐶+72=360𝑚𝐴𝐶+𝑚𝐵𝐶=288.∘∘∘∘

Since 𝑚𝐴𝐶=𝑚𝐵𝐶, we can rewrite this equation as 𝑚𝐵𝐶+𝑚𝐵𝐶=2882𝑚𝐵𝐶=288𝑚𝐵𝐶=144.∘∘∘

Hence, the measure of 𝐵𝐶 is 144∘.

Let’s now apply both theorems simultaneously to solve a problem involving parallel chords and tangents to a circle.

In the following figure, 𝑀 is a circle, 𝐴𝐵 and 𝐶𝐷 are two chords of the circle, and ⃖⃗𝐸𝐹 is a tangent to the circle at 𝐸. If 𝐴𝐵⫽𝐶𝐷⫽⃖⃗𝐸𝐹, the measure of 𝐴𝐶=30∘, and the measure of 𝐷𝐸=74∘, find the measure of 𝐴𝐵.

Answer

Since 𝐴𝐵⫽𝐶𝐷⫽⃖⃗𝐸𝐹, we can apply the theorems of parallel chords and tangents in a circle to find the measure of 𝐴𝐵. That is, the measures of the arcs between parallel chords of a circle are equal and the measures of the arcs between a parallel chord and a tangent of a circle are equal.

We are given 𝑚𝐴𝐶=30∘, so 𝑚𝐵𝐷=30∘ since 𝐴𝐵⫽𝐶𝐷.

Similarly, 𝑚𝐷𝐸=74=𝑚𝐶𝐸∘ since 𝐶𝐷⫽⃖⃗𝐸𝐹.

The sum of the measures of all the arcs that make up the circle is 360∘, so we can form and solve an equation to find 𝑚𝐴𝐵: 𝑚𝐴𝐵+𝑚𝐴𝐶+𝑚𝐶𝐸+𝑚𝐵𝐷+𝑚𝐷𝐸=360𝑚𝐴𝐵+30+74+30+74=360𝑚𝐴𝐵+208=360𝑚𝐴𝐵=152.∘∘∘∘∘∘∘∘∘

In our previous examples, we have applied the theorems of parallel chords and tangents in a circle to find missing values given information about their chords and tangents. These properties can also be applied alongside geometric properties of polygons to help us find missing values. We will demonstrate this in the next example.

In the following figure, a rectangle 𝐴𝐵𝐶𝐷 is inscribed in a circle, where the measure of 𝐴𝐵=71∘. Find the measure of 𝐴𝐷.

Answer

Since 𝐴𝐵𝐶𝐷 is a rectangle, 𝐴𝐵 is parallel to 𝐷𝐶 and 𝐷𝐴 is parallel to 𝐵𝐶. Since these line segments are chords of a circle, we can use the following theorem: the measures of the arcs between parallel chords of a circle are equal.

Since 𝑚𝐴𝐵=71∘, 𝑚𝐶𝐷=71∘. Since the sum of all the arc measures that make up the circle is 360∘, we form and solve the equation as follows: 𝑚𝐴𝐵+𝑚𝐵𝐶+𝑚𝐶𝐷+𝑚𝐴𝐷=36071+𝑚𝐵𝐶+71+𝑚𝐴𝐷=360142+𝑚𝐵𝐶+𝑚𝐴𝐷=360.∘∘∘∘∘∘

Since 𝐴𝐷 is parallel to 𝐵𝐶, 𝑚𝐴𝐷=𝑚𝐵𝐶. Therefore, we can form the following equation: 142+2𝑚𝐴𝐷=3602𝑚𝐴𝐷=218𝑚𝐴𝐷=109.∘∘∘∘

The measure of 𝐴𝐷 is 109∘.

In our final example, we will demonstrate how to apply the theorems of parallel chords and Angle Relationships to solve problems involving algebraic expressions for arc measures.

In the following figure, 𝐴𝐵 and 𝐸𝐹 are two equal chords. 𝐵𝐶 and 𝐹𝐸 are two parallel chords. If the measure of 𝐴𝐶=120∘, find the measure of 𝐶𝐸.

Answer

Firstly, since 𝐴𝐵 and 𝐸𝐹 are equal in length, we can deduce that the measures of their arcs must also be equal. That is, 𝑚𝐴𝐵=𝑚𝐸𝐹=𝑥.∘

Similarly, since 𝐵𝐶 and 𝐹𝐸 are parallel, we know that the measures of the arcs between them are equal. That is, 𝑚𝐶𝐸=𝑚𝐵𝐹=(𝑥+30).∘

Since the sum of all of the arc measures is 360∘, then 𝑚𝐴𝐵+𝑚𝐵𝐹+𝑚𝐸𝐹+𝑚𝐶𝐸+𝑚𝐴𝐶=360𝑥+𝑥+30+𝑥+30+𝑥+120=3604𝑥+180=3604𝑥=180𝑥=45.∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘

Since 𝑚𝐶𝐸=(𝑥+30)∘, we substitute 𝑥=45 into this expression: 𝑚𝐶𝐸=(45+30)=75.∘∘

We will now recap the key concepts from this explainer.

• The measures of the arcs between parallel chords of a circle are equal. Similarly, if the measures of the two arcs between two distinct chords are equal, then the chords are parallel.
• If two chords are equal in length, then the arcs between the endpoints of each chord will be equal in measure.
• The measures of the arcs between a parallel chord and tangent of a circle are equal.