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 .
A chord is a line segment whose endpoints lie on the circumference of the circle. In the diagram, is a chord.
Similarly, a tangent to a circle is a line that intersects the circle exactly once. In the diagram, is a tangent to the circle at point .
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 and . The shorter arc is known as the minor arcΒ (this is the arc whose measure is less than ), and the longer arc (the arc with a measure greater than ) is called the major arc. We denote the minor arc from to more succinctly as .
With these definitions in mind, we will now define and prove a theorem that links parallel chords and arcs in a circle.
Theorem: The Measures of Arcs between Parallel Chords
The measures of the arcs between parallel chords of a circle are equal.
In the diagram, is parallel to , so .
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.
Example 1: Using Theories of Parallel Chords to Find the Measure of an Arc
In the given figure, if the measure of arc , find the measure of arc .
Answer
Recall that arcs formed by a pair of parallel chords are congruent. In the diagram, and are parallel chords, so the arcs formed are congruent. That is, .
Next, since is a chord that passes through the center of the circle, it is a diameter. Hence, the measure of arc is .
By splitting into three separate arcs, we can use this information to form and solve an equation for the measure of arc :
Hence,
In our next example, we will use this theorem alongside properties of angles to find a missing arc measure.
Example 2: Using Theories of Parallel Chords and Angle Relationships to Find the Measure of an Arc
Find where is the center of the circle.
Answer
Recall that arcs formed by a pair of parallel chords are congruent. Since and are parallel, it follows that .
We can see that there is a pair of vertically opposite angles, and . Since vertically opposite angles are equal,
Then, since is a central angle subtended by ,
We know that this is equal to ; hence,
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 and are equal in length, .
In our next example, we will demonstrate how to apply these properties.
Example 3: Using Theories of Parallel Chords and Angle Relationships to Prove Parallel Chords
In the given figure, the measure of ,
the measure of , and the measure of
. What can we conclude about and ?
- They are parallel.
- They are neither parallel nor perpendicular.
- They are perpendicular.
- They are the same length.
- They are parallel and of the same length.
Answer
Letβs 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 .
Hence,
So,
Next, we recall that if the measures of the two arcs between two chords are equal, then the chords must be parallel. Since , then chords and are parallel, and the answer is A or E. We observe that since , 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.
Theorem: The Measures of Arcs between a Parallel Chord and a Tangent
The measures of the arcs between a parallel chord and a tangent of a circle are equal.
In the diagram, is parallel to the tangent at , hence .
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, letβs demonstrate its application.
Example 4: Using Theories of Parallel Chords and Tangent Relationships to Find the Measure of an Arc
is a circle, where is a chord and is a tangent. If and the measure of , find the measure of .
Answer
Since , 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 . We are given that , and we know that the sum of the measures of all the arcs that make up the circle is . Hence,
Since , we can rewrite this equation as
Hence, the measure of is .
Letβs now apply both theorems simultaneously to solve a problem involving parallel chords and tangents to a circle.
Example 5: Using Theories of Parallel Chords and Tangent Relationships to Find the Measure of an Arc
In the following figure, is a circle, and are two chords of the circle, and is a tangent to the circle at . If , the measure of , and the measure of , find the measure of .
Answer
Since , we can apply the theorems of parallel chords and tangents in a circle to find the measure of . 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 , so since .
Similarly, since .
The sum of the measures of all the arcs that make up the circle is , so we can form and solve an equation to find :
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.
Example 6: Using Theories of Parallel Chords and Angle Relationships to Find the Measure of an Arc Using Rectangles
In the following figure, a rectangle is inscribed in a circle, where the measure of . Find the measure of .
Answer
Since is a rectangle, is parallel to and is parallel to . 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 , . Since the sum of all the arc measures that make up the circle is , we form and solve the equation as follows:
Since is parallel to , . Therefore, we can form the following equation:
The measure of is .
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.
Example 7: Using Theories of Parallel Chords and Angle Relationships to Find the Measure of an Arc
In the following figure, and are two equal chords. and are two parallel chords. If the measure of , find the measure of .
Answer
Firstly, since and are equal in length, we can deduce that the measures of their arcs must also be equal. That is,
Similarly, since and are parallel, we know that the measures of the arcs between them are equal. That is,
Since the sum of all of the arc measures is , then
Since , we substitute into this expression:
We will now recap the key concepts from this explainer.
Key Points
- 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.
