Lesson 7 | Circles | 10th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

G.C.A.3 — Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Circles

G.C.A.3 — Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Foundational Standards

The foundational standards covered in this lesson

G.CO.D.13

Congruence

G.CO.D.13 — Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Criteria for Success

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

Use intercepted arcs to show that opposite angles of an inscribed quadrilateral are supplementary. See Unit 7 Glossary for a visual.
Use auxiliary radii to show that opposite angles of an inscribed quadrilateral are supplementary. See Unit 7 Glossary for a visual.
Determine whether a given quadrilateral can be inscribed in a circle.
Find missing angle measures for a quadrilateral inscribed in a circle.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The following resources may be helpful in your planning since it describes more complete classifications of quadrilaterals than typical: Cut the Knot, Alexander Bogomolny, “Classification of Quadrilaterals."
This lesson connects to Unit 1, Lesson 15, which inscribes triangles in circles, by taking that lesson one step further by inscribing quadrilaterals in circles. For an extension, you could inscribe triangles, squares, and regular hexagons in circles.

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Anchor Problems

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

Problem 1

Below is a quadrilateral inscribed in a circle, also known as a cyclic quadrilateral.

Describe the relationship between the opposite angles $$\angle CDB$$ and $$\angle CEB$$ in the cyclic quadrilateral using the intercepted arcs.

Guiding Questions

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References

GeoGebra Geo_U7_L7_AP1

Problem 2

Below is the same cyclic quadrilateral as shown in Anchor Problem #1. Show, using the radii marked and the angle measures marked, the relationship between opposite angles.

Guiding Questions

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References

GeoGebra Geometry - 8.7 AP2

Target Task

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

What value of $$x$$ guarantees that the quadrilateral shown in the diagram below is cyclic? Explain your reasoning.

References

EngageNY Mathematics Geometry > Module 5 > Topic E > Lesson 20 — Exit Ticket, Question #1

Geometry > Module 5 > Topic E > Lesson 20 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Additional Practice

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.

Include a problem where the students are required to use arcs to describe the interior angles of a cyclic triangle.

Open Middle Circles and Square
EngageNY Mathematics Geometry > Module 5 > Topic E > Lesson 20
Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 3, "Cyclic Polygons"
JMAP G.C.A.3: Inscribed Quadrilaterals

Lesson 6

Lesson 8

Lesson Map

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

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

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

G.C.A.2

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

G.C.A.2

Determine the angle and length relationships between intersecting chords.

G.C.A.2

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

G.C.A.3

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

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

G.C.A.2

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

G.C.B.5

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.

G.C.B.5

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

G.C.B.5

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

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

Circles

Objective

Common Core Standards

Core Standards

Circles

Foundational Standards

Congruence

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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