MGSE9-12.G.CO: Congruence

MGSE9-12.G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines

MGSE9-12.G.CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

MGSE9-12.G.CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

MGSE9-12.G.CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

MGSE9-12.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.

Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Translations

MGSE9-12.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. (Extend to include HL and AAS.)

Proving Triangles Congruent

MGSE9-12.G.CO.9: Prove theorems about lines and angles.

Investigating Angle Theorems

MGSE9-12.G.CO.10: Prove theorems about triangles.

Pythagorean Theorem
Triangle Angle Sum
Triangle Inequalities

MGSE9-12.G.CO.11: Prove theorems about parallelograms.

Parallelogram Conditions
Special Parallelograms

MGSE9-12.G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

MGSE9-12.G.SRT: Similarity, Right Triangles, and Trigonometry

MGSE9-12.G.SRT.1a: The dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged.

Dilations

MGSE9-12.G.SRT.1b: The dilation of a line segment is longer or shorter according to the ratio given by the scale factor.

Dilations
Similar Figures

MGSE9-12.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.

Circles
Dilations
Similar Figures
Similarity in Right Triangles

MGSE9-12.G.SRT.4: Prove theorems about triangles.

Pythagorean Theorem
Similar Figures

MGSE9-12.G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Dilations
Perimeters and Areas of Similar Figures
Similarity in Right Triangles

MGSE9-12.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.

Sine, Cosine, and Tangent Ratios

MGSE9-12.G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

MGSE9-12.G.C: Circles

MGSE9-12.G.C.1: Understand that all circles are similar.

Circles

MGSE9-12.G.C.2: Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. 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.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

MGSE9-12.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.

Chords and Arcs

MGSE9-12.G.GPE: Expressing Geometric Properties with Equations

MGSE9-12.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.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

MGSE9-12.G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Distance Formula

MGSE9-12.G.GMD: Geometric Measurement and Dimension

MGSE9-12.G.GMD.1a: Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

MGSE9-12.G.GMD.1b: Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri’s principle.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

MGSE9-12.G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders
Pyramids and Cones

MGSE9-12.S.CP: Conditional Probability and the Rules of Probability

MGSE9-12.S.CP.1: Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not).

Independent and Dependent Events
Probability Simulations
Theoretical and Experimental Probability

MGSE9-12.S.CP.2: Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent.

Independent and Dependent Events

MGSE9-12.S.CP.3: Understand the conditional probability of A given B as P (A and B)/P(B). Interpret independence of A and B in terms of conditional probability; that is, the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Independent and Dependent Events

MGSE9-12.S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in context.

Independent and Dependent Events