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2. Circle Theorems
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Chapter 5
2. 

Circle Theorems

Circles, with their intricate properties, have given rise to several theorems that help in understanding their geometry. Among these, the outside angle theorem sheds light on the relationship between an exterior angle and the circle's arcs. On the other hand, central and inscribed angles play pivotal roles in circle-related problems. A central angle is formed by two radii, while an inscribed angle involves a chord and an arc. These angles and their properties are vital in many real-world applications, from designing circular objects to deciphering complex geometrical patterns.
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17 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Circle Theorems
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An angle can be constructed by the lines and line segments that intersect a circle. In this lesson, the angles related to circles and their properties will be explored.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Explore

Investigating Inscribed Angles of a Circle

Consider a circle with a radius of units. An angle whose sides are two chords of the circle is formed as shown. Move the points and along the circle so that the angle is a right angle.

Think about the following questions.

  • When the measure of the angle is what can the chord joining and be called?
  • Does the arc that the angle intercepts have a special name? If yes, what is it?
Explore

Inscribed and Central Angles of a Circle

On the circle above, construct an angle with a vertex at the center of the circle. The angle being constructed should also cut off the same arc. In other words, construct the corresponding central angle.

Recall that the measure of an arc is the measure of the central angle that includes the arc. Now, move the points and make conjectures that explain the relationship between the measures of angles and arcs.
Discussion

Inscribed Angle of a Circle

In the circles shown previously, the angle formed by two chords is called an inscribed angle.

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides are secant to the circle.
Discussion

Inscribed Angle Theorem

Observe the measures of the angle and the arc intercepted by the angle. Start by moving so that and are collinear. Then, move it once again so that and are collinear.

As can be seen, when and are collinear, becomes the diameter of the circle, and the angle cuts off the semicircle. Furthermore, the measure of is twice the measure of an inscribed angle that intercepts it. This statement can be restated as a theorem.

The measure of an inscribed angle is half the measure of its intercepted arc.

Inscribed Angle

In this figure, the measure of is half the measure of

Proof

An inscribed angle can be created in three different ways depending on where the angle is located in relation to the center of the circle.
Inscribed Angle
Consider Case II. A diameter can be drawn from through dividing the inscribed angle into two.
Inscribed Angle

Let the measures of and be and respectively.

Inscribed Angle

Because the radii of a circle are all congruent, two isosceles triangles can be obtained by drawing and

Inscribed Angle

Therefore, by the Isosceles Triangle Theorem, the measures of and will also be and respectively.

Inscribed Angle

By the Triangle Exterior Angle Theorem, it is recognized that and

Inscribed Angle
The measure of an intercepted arc is the same as the measure of its central angle. Therefore, the measures of and are and respectively.
From here, using the Angle Addition Postulate and the Arc Addition Postulate, and can be written in terms of and
Consequentially, the measure of is half the measure of The proof of Case II has been completed.

By applying similar logic as the procedure above, Case I and Case III can be proven.

Example

Finding the Inscribed Angle of a Circle

In the diagram, the vertex of is on the circle and the sides of the angle are chords of the circle. Given the measure of , find the measure of the angle.

Write the answer without the degree symbol.

Hint

The Inscribed Angle Theorem can be used to find the measure of the angle.

Solution

The angle shown in the diagram fits the definition of an inscribed angle. For this reason, the measure of the angle can be found using the Inscribed Angle Theorem. The theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

The arc intercepted by measures
Substitute

Multiply

Multiply

Therefore, measures
Pop Quiz

Practice Finding Inscribed Angles

Find the measure of the inscribed angle in the circle.

Inscribed Angle
Example

Finding the Central Angle of a Circle

Similarly, given the measure of an inscribed angle, the measure of its corresponding central angle can be found using the Inscribed Angle Theorem. This can be done because the measure of the central angle is the same as the measure of the arc that the central angle cuts off.

In the circle, measures

Find the measure of the corresponding central angle.

Hint

Start by drawing the corresponding central angle.

Solution

Recall that a central angle is an angle whose vertex lies at the center of the circle. Additionally, the inscribed angle and its corresponding central angle intercept the same arc for this example. Therefore, the corresponding central angle is

The arc is included in Therefore, the measure of is equal to the measure of Using the Inscribed Angle Theorem, the measure of can be found.
Substitute

▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

Therefore, also measures
Pop Quiz

Practice Finding Central Angles

Given the measure of an inscribed angle, find the measure of its corresponding central angle.

Inscribed Angle
Discussion

Inscribed Angles of a Circle Theorem

Up to now, the relationship between inscribed angles and their corresponding central angles has been discussed. Now the relationship between two inscribed angles that intercept the same arc will be investigated.

As can be observed, the angles are congruent, so long as they intercept the same arc.

If two inscribed angles of a circle intercept the same arc, then they are congruent.

A circle with two inscribed angles that intercept the same arc

By this theorem, and in the above diagram are congruent angles.

Proof

Consider two inscribed angles and that intercept the same arc in a circle.

Two inscribed angles intercept the same arc
By the Inscribed Angle Theorem, the measures of and are half the measure of
Therefore, by the Transitive Property of Equality, it can be said that these two angles have the same measure.
Consequently, by the definition of congruent angles, and are congruent.

Example

Using the Inscribed Angles of a Circle Theorem

Mark and Jordan have been asked to find the measure of

Mark claims that the measure of is the same as the measure of Jordan, however, thinks that its measure is half the measure of Who is correct?
Find the measure of

Hint

Determine which angles intercept the same arc. Use the Inscribed Angles of a Circle Theorem to find

Solution

The inscribed angles and intercept

By the Inscribed Angles of a Circle Theorem, is congruent to
Accordingly, half the measure of can be found as Additionally, and intercept the same arc,
Again by the Inscribed Angles of a Circle Theorem, is congruent to Therefore,
Since neither Mark's claim nor Jordan's claim is correct.
Discussion

Constructing Angles Outside of a Circle

Inscribed angles, or the central angles, are not the only angles related to circles. In the next part, the angles constructed outside the circles will be examined. To construct an angle outside a circle, tangents can be used.

Concept

Tangent

A tangent is a line, segment, or ray that intersects a circle at exactly one point. The point is called point of tangency, and the line, segment, or ray is said to be tangent to the circle.

Tangent and Point of Tangency
Rule

Tangent to a Circle Theorem

A line is a tangent to a circle if and only if the line is perpendicular to the endpoint of a radius on the circle's circumference.

Circle Q and tangent line m

Based on the diagram, the following relation holds true.

Line is tangent to

Proof

The theorem will be proven in two parts as it is a biconditional statement. Each will be proven by using an indirect proof.

Part I: If a Line Is a Tangent to a Circle, Then It Is Perpendicular to the Endpoint of a Radius on the Circle’s Circumference

Assume that line is tangent to the circle centered at and not perpendicular to By the Perpendicular Postulate, there is another segment from that is perpendicular to Let that segment be The goal is to prove that must be that segment. The following diagram shows the mentioned characteristics.

Line m is perpendicular to segment QT but not perpendicular to segment QP.
Since is said to be perpendicular to is a right triangle. In this case, whose length is units, is the hypotenuse and, therefore, the longest side. As a result, is less than
There is a part of that lies outside of the circle. Let be the length of that part. The other part of is the radius of because it is a segment from the center of the circle to a point on the circle.
Segment QT has a length of b plus r.
By the Segment Addition Postulate, is equal to the sum of and This expression, together with can be substituted into the aforementioned inequality.
SubstituteExpressions

Substitute expressions

SubIneq

Since a length cannot be less than the assumption that is not perpendicular to the line must be false. Therefore, is perpendicular to the tangent at its endpoint on the circle. This concludes the first part of the proof.

Line is tangent to

Part II: If a Line Is Perpendicular to the Endpoint of a Radius on a Circle’s Circumference, Then It Is a Tangent to the Circle

For the second part, it will be assumed that is perpendicular to the radius at and that line is not tangent to In this case, line intersects at a second point

Line m is perpendicular to segment QP but not tangent to the circle.
Since is said to be perpendicular to at is a right triangle and is the hypotenuse. Therefore, is less than
However, it can also be seen that both and are radii of Therefore, they have the same length.
These two statements are contradictory to each other. Therefore, both cannot be true. The contradiction came from supposing that line was not tangent to Consequently, must be a tangent line to the circle. This completes the second part of the proof.

line is tangent to

Having proven both parts, the proof of the biconditional statement of the theorem is now complete.

Line is tangent to

The method used to prove the theorem is called indirect proof.
Example

Using the Tangent to Circle Theorem

In the diagram, is tangent to the circle at the point and is a diameter.

If the measure of is Find the measure of

Hint

A tangent is perpendicular to the radius of a circle through the point of tangency.

Solution

It has been given that By the Tangent to Circle Theorem, is perpendicular to In other words,

Since the sum of the measures of the interior angles of a triangle is
Notice that is an inscribed angle on the diameter of Therefore, measures
Once again, using the Triangle Sum Theorem, the measure of can be found to be
Since is associated with and the measure of the arc can be found using the Inscribed Angle Theorem.
Substitute

▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

Discussion

Circumscribed Angle Theorem

When two tangents of a circle intersect at an exterior point, the angle formed is called a circumscribed angle.

In the diagram above, and are tangents to the circle and is a circumscribed angle.

A circumscribed angle is supplementary to the central angle it cuts off.

The measure of a circumscribed angle is equal to minus the measure of the central angle that intercepts the same arc.

Considering the above diagram, the following relation holds true.

Proof

By definition, a circumscribed angle is an angle whose sides are tangents to a circle. Since is a circumscribed angle, and are tangents to at points and respectively. By the Tangent to Circle Theorem, is perpendicular to and is perpendicular to

Notice that is a quadrilateral and two of its angles are right angles. Recall that the sum of all of the angles in a quadrilateral is Substituting the known angle measures and solving for will give the desired equation.
SubstituteII

,

▼
Solve for
AddTerms

Add terms

SubEqn

SubEqn

This completes the proof.
Pop Quiz

Practice Using the Circumscribed Angle Theorem

Find the measure of the central angle.

Inscribed Angle
Example

Using Circle Theorems to Solve Problems

The following example involving circumscribed angles and inscribed angles could require the use of the previously learned theorems.

Two tangents from to are drawn. The measure of is

Find the measure of the inscribed angle that intercepts the same arc as

Hint

Use the Circumscribed Angle Theorem and Inscribed Angle Theorem.

Solution
The inscribed angle that intercepts is By the Inscribed Angle Theorem, is half
Recall that the measure of an arc is the same as the measure of its corresponding central angle. Therefore, is equal to
Substitute into the above equation.
Now, by the Circumscribed Angle Theorem, is minus
It is given that Substituting that measure into the equation will give the measure of the inscribed angle.
Substitute

▼
Evaluate right-hand side
SubTerm

Subtract term

Multiply

Multiply

Closure

Summarizing Angle Theorems Related to Circles

This lesson defined three angles related to circles as well as the relationships between these angles. The diagram below shows the definitions and the main theorems of this lesson.



Level 1
Level 2
Level 3
Circle Theorems
Exercises

Determine

Circle with a central angle of v and an inscribed angle of 55
Circle with a central angle of 150 and an inscribed angle of v
Circle with a central angle of v and an inscribed angle of 45

The known angle is the inscribed angle to the intercepted arc AB. Additionally, v is the central angle to the same intercepted arc. Since the central angle and its intercepted arc have the same measure, we know that AB equals v.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half that of its intercepted arc.


Like in Part A, we have a central angle and an inscribed angle that both lie on the same intercepted arc. The intercepted arc has the same measure as its central angle.

Again, we can use the Inscribed Angle Theorem to obtain the measure of v.


Like in previous parts, we see that AB is the intercepted arc to both the central angle v and the inscribed angle C. Let's mark the intercepted arc in the diagram.

Once more we will use the Inscribed Angles Theorem to find v.


What is the sum of and

Circle with two inscribed arcs.

The 50^(∘)-angle and u are inscribed angles on the same intercepted arc. Therefore, according to the Inscribed Angles of a Circle Theorem, they are congruent.

Similarly, the 30^(∘)-angle and w are also on the same intercepted arc which means they are congruent.

Now that we know the value of both w and u, we can determine their sum. w+u=30^(∘)+50^(∘)=80^(∘)

Determine

Circle with inscribed angle BOA and two more angles marked

Examining the diagram, we see that the angles ∠ BOC, ∠ AOB, and v together form a straight angle.

Therefore, if we can determine ∠ AOB, we can find m∠ v by subtracting ∠ BOC and ∠ AOB from 180^(∘). Notice that ∠ AOB is the central angle to the intercepted arc BA.

Using the Inscribed Angle Theorem, we can determine the measure of the intercepted arc BA.

Since the central angle and its intercepted arc have the same measure, we know that ∠ AOB equals 60^(∘).

Now we can determine the value of v. 70^(∘)+60^(∘)+v=180^(∘) ⇔ v= 50^(∘)

Determine

Triangle ABC inscribed in a circle.

Examining the diagram, we see that one of the inscribed triangle's sides also is the diameter of the circle. Therefore, the intercepted arc AB has a measure of 180^(∘). According to the Inscribed Angles Theorem, we know that ∠ C must be a right angle.

Since we know two angles in the triangle, we can use the Interior Angles Theorem to determine m∠ x. m∠ A+m∠ B+m∠ C = 180^(∘) If we substitute m∠ A= m∠ x, m∠ B= 50^(∘), and m∠ C= 90^(∘), we can determine m∠ x.

Consider the following circle.

A circle P with several arcs marked.
 Find
Find

We want to determine the measure of the arc BD. Since the arc is described only with the endpoints B and D, it is referring to the minor arc. Let's highlight BD on the circle.

Notice that ∠ BPA and ∠ BPD form a linear pair. This means that the sum of the angle measures is 180^(∘). m∠ BPA+m∠ BPD=180^(∘) If we substitute the value of m∠ BPA in this equation, we can solve for m∠ BPD.

Since the central angle and the intercepted arc has the same measure we know that mBD=139^(∘).

This time we want to determine mABC. Since the arc is described with the endpoints A and C and a point B on the arc, it is referring to the arc we can draw between A and C that also goes through B.

From Part A, we know that mABD has a measure of 180^(∘). Additionally, ∠ DPC and ∠ APB make a pair of vertical angles. Therefore, they are congruent according to the Vertical Angles Theorem. mDPC=41^(∘) With this information, we can determine the measure of ABD. mABD=180^(∘)+41^(∘)=221^(∘)

Circle Theorems

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