**CONGRUENCE (G. CO)**

__UNDERSTAND CONGRUENCE IN TERMS OF RIGID MOTIONS__**MCC9‐12.G.CO.6**Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

**MCC9‐12.G.CO.7**Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

**MCC9‐12.G.CO.8**Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

**PROVE GEOMETRIC THEOREMS****MCC9‐12.G.CO.9**Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

**MCC9‐12.G.CO.10**Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

**MCC9‐12.G.CO.11**Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

**MAKE GEOMETRIC CONSTRUCTIONS****MCC9‐12.G.CO.12**Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

**MCC9‐12.G.CO.13**Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle

**SIMILARITY, TRIGHT TRIANGLES, AND TRIGONOMETRY (G.SRT)**

__UNDERSTANDING SIMILARITY IN TERMS OF SIMILARITY TRANSFORMATIONS__**MCC9‐12.G.SRT.1**Verify experimentally the properties of dilations given by a center and a scale factor:

- a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
- b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

**MCC9‐12.G.SRT.2**Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

**MCC9‐12.G.SRT.3**Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

__PROVING THEOREMS INVOLVING SIMILARITY__**MCC9‐12.G.SRT.4**Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

**MCC9‐12.G.SRT.5**Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

__DEFINE TRIGONOMETRIC RATIOS AND SOLVE PROBLEMS INVOLVING RIGHT TRIANGLES__**MCC9-12.G.SRT.6**Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

**MCC9-12.G.SRT.7**Explain and use the relationship between the sine and cosine of complementary angles.

**MCC9-12.G.SRT.8**Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

**CIRCLES (G.C)**

**UNDERSTAND AND APPLY THEOREMS ABOUT CIRCLES****MCC9-12.G.C.1**Prove that all circles are similar.

**MCC9-12.G.C.2**Identify and describe relationships among inscribed angles, radii, and chords. Include 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.

**MCC9-12.G.C.3**Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

**MCC9-12.G.C.4 (+)**Construct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circles

**MCC9-12.G.C.5**Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

- Use the properties of circles to solve problems involving the length of an arc and the area of a sector.

**EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS (G.CPE)**

__TRANSLATE BETWEEN THE GEOMETRIC DESCRIPTION AND THE EQUATION FOR A CONIC SECTION__**MCC9-12.G.GPE.1**Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

- Translate between the geometric description and the equation for a circle

**MCC9-12.G.GPE.2**Derive the equation of a parabola given a focus and directrix. Use coordinates to prove simple geometric theorems algebraically

__USE COORDINATES TO PROVE SIMPLE GEOMETRIC THEOREMS ALGEBRICALLY__**MCC9-12.G.GPE.4**Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

**GEOMETRIC MEASUREMENT AND DIMENSION (G.GMD)**

__EXPLAIN VOLUME FORMULAS AND USE THEM TO SOLVE PROBLEMS__**MCC9-12.G.GMD.1**Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

**MCC9-12.G.GMD.2**(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

**MCC9-12.G.GMD.3**Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

**Navigation**

Geometry

**Congruence (G.CO) :****MCC9‐12.G.CO.6 |**

Similarity, Right Triangles & Triangles (G.SRT)

Circles (G.C)

Expressing Geometric Properties with Equations (G.CPE)

Geometric Measurement and Dimension (G.GMD)Similarity, Right Triangles & Triangles (G.SRT)

Circles (G.C)

Expressing Geometric Properties with Equations (G.CPE)

Geometric Measurement and Dimension (G.GMD)