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 different cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure.

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 inscribed angle is half the angle measure of its intercepted arc.
Explain and understand that all circles are similar.

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

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 given point.
Students prove that tangent segments from the same point are equal in length.

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 given point.
Students recognize and use the fact if a circle is tangent to both rays of an angle, then its center lies on the angle bisector.

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 is tangent).
Students solve a variety of missing angle problems using the inscribed angle theorem.

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 lines of the circle.
Students discover that the measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

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.

Students write the equation for a circle in center-radius form, (x - a)2 (y - b)2 = r2 using the Pythagorean theorem or the distance formula.
Students write the equation of a circle given the center and radius. Students identify the center and radius of a circle given the equation.

Students complete the square in order to write the equation of a circle in center-radius form.
Students recognize when a quadratic in x and y is the equation for a circle.

Given a circle, students find the equations of two lines tangent to the circle with specified slopes.
Given a circle and a point outside the circle, students find the equation of the line tangent to the circle from that point.

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) where w is the measure of the acute angle formed by diagonals AB and CD.

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 that for a cyclic quadrilateral ABCD, AC·BD = AB·CD + BC·AD. They explore applications of the result.

The purpose of this task is to solidify understanding of the equation of the circle. The task begins with sketching circles and writing their equations. Students are challenged to reverse the process to find the center of the circle.

The purpose of this task is for students to find roots of polynomials and write the polynomials in factored form. This task builds on previous algebraic work, including factoring, polynomial long division, and quadratic formula. Students also use their knowledge of the Fundamental Theorem of Algebra to know how many roots a function has and to consider the possible combinations of real and imaginary roots for polynomials of degree 1-4. Students learn that imaginary roots occur in conjugate pairs and use this knowledge to both find roots and know the possible combinations of roots for polynomials.

This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The final task assesses student mastery of the geometry standards related to finding the volume of geometric figures.

The Common Core State Standards (CCSS) videos are designed to support states, schools, and teachers in the implementation of selected math standards. Each video is an audiovisual resource that focuses on one or more specific standards through examples and illustrations geared to enhancing understanding. The intent of each content-focused video is to clarify the meaning of the individual standard rather than to be a guide on how to teach each standard, although the examples can be adapted for instructional use.

The Common Core State Standards (CCSS) videos are designed to support states, schools, and teachers in the implementation of selected math standards. Each video is an audiovisual resource that focuses on one or more specific standards through examples and illustrations geared to enhancing understanding. The intent of each content-focused video is to clarify the meaning of the individual standard rather than to be a guide on how to teach each standard, although the examples can be adapted for instructional use.