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Geometry Module 5, Topic A: Central and Inscribed Angles

Classroom Resource Information

Title:

Geometry Module 5, Topic A: Central and Inscribed Angles

URL:

https://www.engageny.org/resource/geometry-module-5-topic-a-overview

Content Source:

EngageNY

Type:

Lesson/Unit Plan

Overview:

Module 5, Topic A leads students first to Thales' theorem (an angle drawn from a diameter of a circle to a point on the circle is sure to be a right angle), then to possible converses of Thales' theorem, and finally to the general inscribed-central angle theorem. Students use this result to solve unknown-angle problems. Through this work, students construct triangles and rectangles inscribed in circles and study their properties.

Content Standard(s):

Mathematics
MA2015 (2016)
Grade: 9-12
Geometry

26 ) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis

37. Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: 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.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given circles with two points on the circle,

Compare the measures of the angles (with and without technology) formed by creating radii to the given points, creating chords from a third point on the circle to the given points, and creating tangents from a third point outside the circle to the given points, and conjecture about possible relationships among the angles.
Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle is one half the central angle cutting off the same arc, and the circumscribed angle cutting off that arc is supplementary to the central angle relating all three).

Given circles with chords from a point on the circle to the endpoints of a diameter,

Find the measure of the angles (with and without technology), conjecture about and explain possible relationships.
Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle on a diameter is a right angle).

Given a circle with a tangent and radius intersecting at a point on the circle,

Find the measure of the angle at the intersection point (with and without technology), conjecture about and explain possible relationships.
Use logical reasoning to justify (or deny) the conjectures (in particular justify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Teacher Vocabulary:

Central angles
Inscribed angles
Circumscribed angles
Chord
Circumscribed
Tangent
Perpendicular arc

Knowledge:

Students know:

Definitions and characteristics of central, inscribed, and circumscribed angles in a circle.
Techniques to find measures of angles including using technology (dynamic geometry software).

Skills:

Students are able to:

Explain and justify possible relationships among central, inscribed, and circumscribed angles sharing intersection points on the circle.
Accurately find measures of angles (including using technology (dynamic geometry software)) formed from inscribed angles, radii, chords, central angles, circumscribed angles, and tangents.

Understanding:

Students understand that:

Relationships that exist among inscribed angles, radii, and chords may be used to find the measures of other angles when appropriate conditions are given.
Identifying and justifying relationships exist in geometric figures.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
GEO.37.1: Define inscribed angles, central angles, circumscribed angles, radius, chord, tangent, secant, and diameter.
GEO.37.2: Define inscribed and circumscribed circle of a triangle.
GEO.37.3: Apply knowledge of arcs, angles and chords to solve circle related problems.
GEO.37.4: Determine angle values for all angles formed in the exterior, interior and on the circle.
GEO.37.5: Determine lengths of intersecting chords and secants.
GEO.37.6: Discuss the relationship among inscribed angles, radii, and chords.
GEO.37.7: Illustrate inscribed and circumscribed circles of a triangle and quadrilaterals inscribed in a circle.
GEO.37.8: Illustrate radii, chords, diameters, tangents to curve, central, inscribed, and circumscribed angles.

Prior Knowledge Skills:

Identify parts of a circle.
Recall how to find circumference of a circle.
Recall the meaning of a radius and diameter.
Identify all types of angles.
Recognize the attributes of a circle.
Identify and label parts of a circle.
Define diameter, radius, circumference, area of a circle, and formula.

Alabama Alternate Achievement Standards

AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.

Tags:

angles, chords, circle, circumscribed, diameter, Inscribed angles, quadrilateral, radii, radius, triangle

License Type:

Custom Permission Type
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Accessibility

Text Resources: Content is organized under headings and subheadings

Comments

There are six lessons on this topic.

This resource is free for teachers to access and use. All resources required for the lessons are available to print from the site.

This resource provided by:

Author:

Hannah Bradley

ALEX Classroom Resource

Geometry Module 5, Topic A: Central and Inscribed Angles