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 Geometry Module 5: Circles With and Without Coordinates This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied...

 This module revisits trigonometry that was introduced in Geometry and Algebra II, uniting and further expanding the ideas of right triangle trigonometry and the unit circle. New tools are introduced...

 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...

 Student Outcomes Using observations from a pushing puzzle, explore the converse of Thales' theorem: If triangle ABC is a right triangle, then A, B, and C are three distinct points on a circle with...

 Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle.

 Student Outcomes Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter.

 Student Outcomes Explore the relationship between inscribed angles and central angles and their intercepted arcs.

 Student Outcomes Prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Recognize and use...

 Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central angles.

 Topic B defines the measure of an arc and establishes results relating chord lengths and the measures of the arcs they subtend. Students build on their knowledge of circles from Module 2 and prove...

 Student Outcomes Define the angle measure of arcs, and understand that arcs of equal angle measure are similar. Restate and understand the inscribed angle theorem in terms of arcs: The measure of an...

 Student Outcomes Congruent chords have congruent arcs, and the converse is true. Arcs between parallel chords are congruent.

 Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a...

 Student Outcomes Students apply their understanding of arc length and area of sectors to solve problems of unknown area and length.

 In Topic C, students explore geometric relations in diagrams of two secant lines, or a secant and tangent line (possibly even two tangent lines), meeting a point inside or outside of a circle. They...

 Student Outcomes Students discover that a line is tangent to a circle at a given point if it is perpendicular to the radius drawn to that point. Students construct tangents to a circle through a...

 Student Outcomes Students use tangent segments and radii of circles to conjecture and prove geometric statements, especially those that rely on the congruency of tangent segments to a circle from a...

 Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays...

 Student Outcomes Students understand that an angle whose vertex lies in the interior of a circle intersects the circle in two points and that the edges of the angles are contained within two secant...

 Student Outcomes Students find the measures of angle/arcs and chords in figures that include two secant lines meeting outside a circle, where the measures must be inferred from other data.

 Student Outcomes Students find “missing lengths” in circlesecant or circlesecanttangent diagrams.

 The module concludes with Topic E focusing on the properties of quadrilaterals inscribed in circles and establishing Ptolemy's theorem. This result codifies the Pythagorean theorem, curious facts...

 Student Outcomes Students show that a quadrilateral is cyclic if and only if its opposite angles are supplementary. Students derive and apply the area of cyclic quadrilateral ABCD as 1/2 AB·CD·sin(w...

 Student Outcomes Students determine the area of a cyclic quadrilateral as a function of its side lengths and the acute angle formed by its diagonals. Students prove Ptolemy’s theorem, which states...