lesson 11 5 problem solving angle relationships in circles

Curriculum  /  Math  /  10th Grade  /  Unit 7: Circles  /  Lesson 10

Lesson 10 of 14

Criteria for Success

Tips for teachers, anchor problems.

• Problem Set

Target Task

Additional practice.

Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems. 

Common Core Standards

Core standards.

The core standards covered in this lesson

G.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.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Identify and use angle and segment relationships in circle diagrams to find missing measures. 
• State and describe the conjectures and theorems used to find missing measures in circle diagrams. 

Suggestions for teachers to help them teach this lesson

This lesson summarizes key concepts in the unit; therefore, this lesson should be used as a review day. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

In circle $$A$$ below, $$\overleftrightarrow{CE}$$ and $$\overleftrightarrow{DE}$$  are tangent at points  $$C$$ and $$D$$ , respectively.

Find $$m\angle{CED}$$  and  $$m\angle{ADG}$$ .

Guiding Questions

Geometry - 8.10 AP1 by Match Fishtank is made available by GeoGebra  under the CC BY-NC-SA 3.0  license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 11:55 a.m..

In the following diagram, the radius of circle  $$D$$  is $$5$$  cm and  $$F$$  is the midpoint of  $$\overline{AE}$$ . The measures of  $$\widehat{EB}$$  and  $$\widehat{BC}$$  are given in the diagram. Find the measures of all other unmarked angles, arcs, and segments.

Module 7: Circles a Geometric Perspective from  Geometry: A Learning Cycle Approach  made available by Mathematics Vision Project  under the CC BY 4.0  license. © 2016 Mathematics Vision Project. Accessed Oct. 19, 2017, 2:58 p.m..

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

In the diagram below,  $$\overline{PA}$$ is tangent to circle $$O$$ , and  $$\overline{AB}$$ is a chord. If $$m\widehat{ACB}=300$$ , find the measure of $$\angle BAP$$ .

G.C.A.2: Chords, Secants, and Tangents 17 is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017  JMAP, Inc. - All rights reserved. Accessed Sept. 19, 2018, 10:43 a.m..

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• EngageNY Mathematics Geometry > Module 5 > Topic C > Lesson 16
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 5, "From Polygons to Circles"
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 6, "Circular Reasoning"
• MARS Formative Assessment Lessons for High School Solving Problems with Circles and Triangles
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 3, "Cyclic Polygons"

Topic A: Equations of Circles

Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.

G.GPE.A.1 G.GPE.B.4

Given a circle with a center translated from the origin, write the equation of the circle and describe its features. 

G.C.A.1 G.CO.A.5 G.GPE.A.1

Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle. 

G.CO.A.5 G.GPE.A.1

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Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem. 

Describe the relationship between inscribed and central angles in terms of their intercepted arc. 

Determine the angle and length relationships between intersecting chords. 

Prove properties of angles in a quadrilateral inscribed in a circle.

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles. 

G.C.A.2 G.C.A.3

Construct tangent lines to a circle to define and describe the circumscribed angle. 

G.C.A.2 G.C.A.4

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians. 

Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius. 

Use sector area of circles to calculate the composite area of figures.

G.C.B.5 N.Q.A.2 N.Q.A.3

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• > Geometry
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Mastering Angles in a Circle: Formulas, Rules, and Applications Unlock the secrets of circular geometry! Learn to identify, calculate, and apply various angles in circles. Master central angles, inscribed angles, and chord properties with our expert guidance.

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• Circumference
• Central angle
• Inscribed angle
• perpendicular bisector
• point of tangency
• What are Inscribed angles and Central angles?

• You are setting up a display board for the science fair. You have two spotlights: one projects through an angle of 20°; and one projects through an angle of 40°. Where is the best place to put the display board? Show your answer in a diagram.

• ∠ \angle ∠ BCD
• ∠ \angle ∠ AEB
• ∠ \angle ∠ CED

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What is a central angle

Previously, when you worked with angles in a triangle, you were mainly working with central angles . These are angles that had their vertices at the center of the circle. However, in this lesson, we won't be working with central angles.

What is an inscribed angle

If we're not working with central angles, then what angles in a circle will we be focusing on? We'll be tackling inscribed angles .

Inscribed angles are angles that can be anywhere on the circle's circumference. While we're not solely working with central angles anymore, you'll learn that the central angle of a circle will still come into play when we're going through the example problems in this lesson.

Inscribed angle theorem

There are several inscribed angle theorems that we'll be learning today. One of them is the angle in the center theorem. This tells us that an inscribed angle θ \theta θ is half of the central angle 2?. This is seen illustrated below.

Angles subtended on the same arc theorem

Another theorem that we'll learn is the angles subtended by the same arc theorem. Given that the endpoints are the same, you'll realize that from the below illustration, it doesn't matter where the inscribed angle θ \theta θ is. It will remain the same. As you can also see, this is because they share the same arc (the orange line), hence the name of this theorem.

In the following diagram, the radius is 24 cm and angle BDC is 75°.

a) Find angle BAC.

Angle BDC is the central angle, which equals 75 degrees. Angle BAC is the inscribed angle. According to the inscribed angle theorem, angle BAC equals half of angle BDC. Conversely, you can think about the relationship between angle BDC and angle BAC as the central angle always being doubled that of the inscribed angle.

Central angle = inscribed angle x 2

75 75 75 ° = 2 θ =2\theta = 2 θ

θ = 37.5 \theta=37.5 θ = 37.5 °

b) Find the chord BC

Let us draw a bisector in triangle DBC to cut it in half

Now, look at one of the halves cut from triangle DBC

X is half of the chord BC that we are looking for. So solve for x using sin (remember the principles of SohCahToa?)

B C = 2 x BC=2x BC = 2 x

sin ⁡ 37.5 \sin 37.5 sin 37.5 ° = x 24 =\frac{x}{24} = 24 x ​

x ≅ 14.6 x \cong 14.6 x ≅ 14.6

B C = 2 x = 29.2 c m BC=2x=29.2cm BC = 2 x = 29.2 c m

Given angle BAE = 44.5° and angle ADC = 64.27°.

Find angle BCD.

Angle BCD and angle BAE are inscribed angles on the same arc. So, angle BCD = BAE = 44.5°. This is based off the angles subtended by the same arc theorem.

Still not sure about the theorems? This online demonstration can show you the proof through you dragging the lines in the circle that form central and inscribed angles.

Next up, you'll be dealing with problems that'll require you to find the arcs of a circle and the area of a sector in circles . You'll also expand on the central and inscribed angles that you learned here, as well as move on to doing proofs on the inscribed angles .

• Central angles and proofs
• Inscribed angles and proofs
• Arcs of a circle
• Areas and sectors of circles
• Central and inscribed angles in circles
• Circle chord, tangent, and inscribed angles proofs

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ANGLE RELATIONSHIPS IN CIRCLES

Using tangents and chords.

We know that the measure of an angle inscribed in a circle is half the measure of its intercepted arc.  This is true even if one side of the angle is tangent to the circle. 

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

In the diagram, show above, we have 

m ∠1  =  1/2  ⋅  m ∠arc AB

m∠2  =  1/2  ⋅  m∠arc BCA

Finding Angle and Arc Measures

Example 1 :

Line m is tangent to the circle. Find the measure of the red angle.

m∠1  =  1/2  ⋅ 150 °

m∠1  =  7 5 °

Example 2 : 

Line m is tangent to the circle. Find the measure of the red arc.

m∠arc RSP  =  2  ⋅ 130 °

m∠arc RSP  =  26 0 °

Finding an Angle Measure

Example 3 :

In the diagram below, BC is tangent to the circle.  Find m ∠ CBD.

m∠ CBD  =  1/2  ⋅  m∠arc DAB

5x  =   1/2  ⋅ (9x + 20)

Multiply each side by 2.

10x  =  9x + 20

Subtract 9x from each side.

x  =  20

So, the angle measure CBD is

m∠ CBD  =  5(2 0 °)

m∠ CBD  =  10 0 °

Lines Intersecting Inside or Outside a Circle

If two lines intersect a circle, there are three places where the lines can intersect.

We know how to find angle and are measures when lines intersect on the circle.  We can use the following Theorems find measures when the lines intersect inside or outside the circle. 

Theorem 1 :

If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

In the diagram shown above, we have

m ∠1  =  1/2  ⋅ ( m ∠arc CD + m ∠arc AB)

m∠2  =  1/2  ⋅ ( m∠arc BC + m∠arc AD)

Theorem 2 :

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

m ∠1  =  1/2  ⋅ ( m ∠arc BC - m ∠arc AC )

m ∠2  =  1/2  ⋅ ( m ∠arc PQR - m ∠arc PR )

m ∠3  =  1/2  ⋅ ( m ∠arc XY - m ∠arc WZ )

Finding the Measure of an Angle Formed by Two Chords

Example 4 :

Find the value of x in the diagram shown below. 

x °  =  1/2  ⋅ ( m ∠arc PS + m ∠arc RQ)

x °  =  1/2  ⋅ (106 °  + 174 ° )

x °  =  1/2  ⋅ (280 ° )

x °  =  140 °

Hence, the value of x is 140. 

Using Theorem 2

Example 5 :

Using Theorem 2, we have

m ∠ GHF  =  1/2  ⋅ ( m ∠EDG - m ∠GF)

72 °  =  1/2  ⋅ (200 ° - x °)

Multiply each side by 2. 

144   =  200  - x

Solve for x.

x  =  56

Example 6 :

Arcs MN and MLN make a whole circle. So, we have

m ∠arc MLN +  m ∠MN   =  360 °

Plug  m ∠MN  =  92 °.

m ∠arc MLN + 92 °   =  360 °

Subtract 92 ° from each side. 

m ∠arc MLN  =  360 ° - 92 °

m ∠arc MLN  =  268 °

x °   =  1/2  ⋅ ( m ∠MLN - m ∠MN)

x °   =  1/2  ⋅ (268 °  - 92 ° )

x  =  1/2  ⋅ (176 )

x  =  88

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