Discover, solve and prove angular relationships when angles are inscribed in circles Introduction to the process of mathematical deductive reasoning leading to formal proofs
Conceptual Framework
logic measurement, generalization and justification in the context of Identities and Relationships
Statements of Inquiry
Understanding form and generalizing geometric relationships allows us to solve real-life problems
Factual
What are the relationships between the angles formed when parallel lines are cut by a transversal?
What is the relationship between angles inscribed in circles?
Identify the relationships formed by the resulting segments when two chords intersect?
Conceptual
How can theorems about circles be justified using concepts of similarity and congruence?
Debatable
How can circle geometry help us to find solutions to real-life problems?
Description
The focus of this unit will be on investigating patterns in mathematics. Throughout this unit, students will be asked to recognize patterns, describe patterns with general rules, verify patterns using a digital tool and begin to provide formal justifications or proofs of these patterns. This unit provides students with an opportunity to apply many of the angle relationships they have studied in previous years and extend these relationships to angles formed by chords, segments and tangents in circles. We will work with old fashioned compasses and protractors to further develop skills in constructing circles, measuring angles and deducing relationships which are later verified in a digital tool such as Geogebra. Students will learn how to use deductive logic to begin to write formal justifications of their resulting generalizations of the circle theorems studied in this unit. The dynamic nature of Geogebra allows students to continually observe these relationships for a large number of special cases.
Learning Outcomes
Define the terms radius, diameter, centre, chord, tangent, semicircle, concentric circles, inscribed angles, central angles, major and minor arcs, segment, sector, tangent, secant, subtended
Review and apply the following angle theorems: angles on a straight line, supplementary angles, complementary angles, vertically opposite angles, angles in a triangle, angles in a quadrilateral, angles around a point, exterior angles
Review and apply the angle relationships formed when a transversal cuts two or more parallel lines
State, prove and apply the following theorems about angles in a circle: angles inscribed in a semicircle, angle at the centre is twice the angle at the circumference, angles inscribed in the same segment, angles in a cyclic quadrilateral, alternate segment theorem
State, prove and apply the following theorems about chords in a circle: perpendicular bisector of a chord passes through the centre (and the converse), intersecting chords theorem
State, prove and apply the following theorems about tangents to a circle: radius drawn to a tangent at the point of tangency, tangents drawn from a point outside the circle