Circle theorems

Here you will learn about circle theorems, including their application, proof, and how to use them to problem solve. You will learn how these theorems apply to angles formed by secant and tangent lines and how they apply to properties within a circle.

Students first learn about circle theorems in geometry where they learn how to apply the theorems to problem solve. Students use the concepts taught in precalculus and calculus as well as on standardized tests such as SATs and state assessments.

What are circle theorems?

Circle theorems verify properties that show relationships between angles formed by special lines and line segments and arcs within a circle. These theorems along with prior knowledge of other angle properties are used together to problem solve.

Let’s take a look in greater depth at the following:

• Central angles
• Chords of a circle
• Circle and chord theorems
• Subtended angles
• Tangents of a circle

Below is a summary of each theorem, along with a diagram. You need to remember all of these circle theorems and be able to describe each one in a sentence.

Central angles

Central angles are angles that have a vertex at the center of the circle and the sides of the angle are the radii of the circle.

Angle ABC is a central angle of circle B.

The degree measurement of a central angle is equal to the arc it intercepts.

Angle ABC=60^{\circ} and arc AC=60^{\circ}

Inscribed angles are angles that have a vertex on the edge of the circle and the sides are chords of the circle. The measurement of the inscribed angle is \cfrac{1}{2} the measurement of the arc it intercepts.

Angle CEA is an inscribed angle (subtended angle) and the measurement of this inscribed angle is 30^{\circ} because it intercepts an arc that is 60^{\circ}.

The inscribed angle is always \cfrac{1}{2} the measurement of the arc it intercepts.

Remember: A subtended angle is an angle inside a circle that is created by two chords that meet at a point on the circle.

Step-by-step guide: Central Angle

Chords of a circle

Chords are line segments inside a circle that have endpoints on the edge of the circle. The longest chord in a circle is the diameter. Chords create situations within a circle that can help us to solve problems.

1. Chords that are congruent intercept congruent arcs. Chord HG and Chord KJ are congruent. So, the arcs they intercept are congruent, Arc HG is congruent to Arc KJ.
   
   
   
2. When the diameter (or radius) is perpendicular to a chord, it also bisects (cuts in half) the chord. Diameter DK in circle C is perpendicular to chord GH, therefore, DK also bisects GH, \, JH \approx GJ.
   
   
   
3. In a circle or congruent circles, two chords are congruent if and only if they are equidistant (the same distance) from the center of the circle.
   
   In circle B, you can determine that chord LM and chord NQ are congruent if you know that they are the same distance (perpendicular distance) from point B.
   
   

If BO is perpendicular to LM and if BP is perpendicular to NQ and BO is congruent in length to BP, you can conclude that LM is congruent to NQ.

Step-by-step guide: Chords of a circle

Step-by-step guide: Circle chord theorems

Subtended angles

Subtended angles of a circle form inscribed angles. When inscribed angles intercept the same arc they are congruent.

Angle KGM, angle KEM and angle KLM all intercept the same arc, arc KM. If arc KM is 60^{\circ}, then angle KGM=30^{\circ}, angle KEM=30^{\circ} and angle KLM=30^{\circ}.

So, \text{ angle } KGM \approx \text { angle } KEM \approx \text{ angle } KLM.

Subtended angles can form inscribed quadrilaterals and triangles.

A quadrilateral can be inscribed in a circle, meaning that all four vertices are on the edge of the circle, if the opposite angles of a quadrilateral are supplementary (sum to 180^{\circ} ).

Inscribed quadrilaterals are also called cyclic quadrilaterals.

Quadrilateral HIJG is inscribed in circle F , so opposite angles must be supplementary.

Inscribed triangle ABC has one side as the diameter of the circle D.

Triangle ABC is a right triangle because the intercepted arc of angle A is semi-circle CB which is 180^{\circ}.

The measure of the intercepted arc is half the measure of the arc it intercepts, so in this case, angle A=90^{\circ}

This is also known as the semicircle theorem.

Step-by-step guide: Subtended angles

Tangents of a circle

Tangent lines are lines that touch the circle at one point called the point of tangency.

Line RS is tangent to circle T if and only if the line is perpendicular to the radius at point W. Point W is the point of tangency.

If two lines are tangent to a circle from an external point, the tangent lines will be congruent in length. The tangent line must be perpendicular to the radius at the point of tangency, so using congruent triangles, you can prove that the tangent lines are congruent.

Through HL (Hypotenuse Leg) you can prove that triangle STR is congruent to triangle WTR.

Therefore, by CPTPC (Corresponding parts of congruent triangles are congruent) you can conclude that segment SR is congruent to segment WR, which are the tangent segments.

Step-by-step guide: Tangents of a circle

What are circle theorems?

Common Core State Standards

How does this apply to geometry?

• High School Geometry – Circles (HSG-C.A.2)
  Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

How to use circle theorems to solve problems

There are several ways to apply circle theorems to problem solve. For more specific step-by-step guides, check out the pages linked in the “What are circle theorems?” section above or read through the examples below.

In order to problem solve using circle theorems:

1. Recall the theorem.
2. Solve the problem.

Circle theorem examples

Example 1: central angle theorem

Find the measure of angle A in circle D.

1. Recall the theorem.

Angle A is an inscribed angle and is half the measure of the arc it intercepts.

2Solve the problem.

Arc PC is the intercepted arc of angle A. Arc PC=42^{\circ} and angle A is an inscribed angle.

Angle A=21^{\circ}

Example 2:

Find the measure of arc WV in circle R.

Example 3:

Find the value of x.

Example 4:

In circle E, find the value of angle G.

Example 5:

Triangle TQR is inscribed in circle P and TQ=QR=10. Find the length of the diameter, TR.

Example 6:

In circle C, the diameter is 14 units and JK is 3 units. Find the length of GH.

Teaching tips for circle theorems

• Provide opportunity for students to discover the theorems and make conjectures using compasses and protractors.

• Use digital platforms such as Desmos so students can investigate strategies to problem solve using the circle theorems.

• Have students use online platforms such as Khan Academy which contain tutorials created by math tutors to practice exam questions and to prepare for the SAT.

• Although worksheets provide an opportunity for students to practice skills, using digital platforms that focus on gameplay is more effective in engaging students.

Easy mistakes to make

• Doubling instead of halving
  Thinking that you should double the intercepted arc measurement to find the measure of the inscribed angle instead of halving it.
  
  For example, thinking that angle CBA is 104^{\circ} instead of 26^{\circ}. Angle CBA=\cfrac{1}{2} \times 52=26
  
  

• Thinking that consecutive angles of an inscribed quadrilateral are always supplementary
  The opposite angles of an inscribed quadrilateral are always supplementary. The consecutive angles may be supplementary but are not always supplementary like in the figures below.
  
  In both circles, the opposite angles must be supplementary because the quadrilaterals are inscribed so. In circle G, the consecutive angles are also supplementary because quadrilateral ABCD is a square.
  
  

• Thinking that any chord that is perpendicular to another chord bisects it
  When the diameter is perpendicular to a chord it will always bisect that chord. But not all chords that are perpendicular bisect each other.
  
  
  In circle C, \, ED is the diameter which is perpendicular to chord FG. \, FG is also being bisected by ED. In circle B, chords HK and IJ are perpendicular but neither one is being bisected.

• Not using triangles to find missing lengths
  For example, in the figure below depending on the given information, it might be helpful to sketch an isosceles triangle because segments connecting CG and CH are radii of the circle and equal in length.
  
  

• Thinking that any polygon inside a circle is inscribed
  For example, looking at the figures below, the quadrilateral inside circle A is not inscribed because the vertices of the quadrilateral are not all on the edge of the circle.
  
  The quadrilateral in circle B is inscribed because all the vertices are on the edge of the circle.
  
  

The next lessons are

• Types of data
• Averages and range
• Representing data

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