High School Geometry (Colorado Academic Standards)

Standard 4: Geometry

• MA.HS.G-CO.A Experiment with transformations in the plane.
  • 1. Know precise definitions of angle, circle, perpendicular and parallel lines and segment from undefined terms.
  • 2. Represent transformations as functions on points and compare rigid motions with others such as stretches.
  • 3. Describe the rotations and reflections that map a rectangle, parallelogram, trapezoid or regular polygon onto itself.
  • 4. Define rotations, reflections and translations using angles, circles, perpendicular and parallel lines and segments.
  • 5. Draw the image of a figure under a rotation, reflection or translation and specify sequences of moves.
• MA.HS.G-CO.B Understand congruence through rigid motions.
  • 1. Use rigid motions to transform figures, predict effects and decide whether two figures are congruent.
  • 2. Use rigid motions to show triangles are congruent exactly when corresponding sides and angles are congruent.
  • 3. Explain how ASA, SAS and SSS follow from congruence defined by rigid motions.
• MA.HS.G-CO.C Prove geometric theorems.
  • 1. Prove theorems about lines and angles (vertical angles, angles with parallel lines, perpendicular bisector points).
  • 2. Prove theorems about triangles (angle sum, isosceles base angles, midsegment, medians meeting at a point).
  • 3. Prove theorems about parallelograms (opposite sides and angles, bisecting diagonals, rectangles' diagonals).
• MA.HS.G-CO.D Make geometric constructions.
  • 1. Make formal constructions with compass and straightedge, paper folding or software (copying, bisecting, perpendiculars, parallels).
  • 2. (+) Construct an equilateral triangle, square and regular hexagon inscribed in a circle. (being checked)
• MA.HS.G-SRT.A Understand similarity through similarity transformations.
  • 1. Check by experiment the properties of dilations with a center and scale factor.
  • 1.a. A dilation maps a line not through the center to a parallel line and leaves lines through the center unchanged.
  • 1.b. A dilated segment is longer or shorter by the scale factor.
  • 2. Use similarity transformations to decide if figures are similar and explain similarity of triangles through matching angles and proportional sides.
  • 3. Use similarity transformations to establish the AA similarity criterion.
• MA.HS.G-SRT.B Prove theorems about similarity.
  • 1. Prove triangle theorems such as the side-splitter theorem and the Pythagorean Theorem using similarity.
  • 2. Use congruence and similarity criteria to solve problems and prove relationships in figures.
• MA.HS.G-SRT.C Define trigonometric ratios and solve right-triangle problems.
  • 1. Understand that side ratios in right triangles depend on the angles, leading to the trigonometric ratios.
  • 2. Explain and use how sine and cosine of complementary angles are related.
  • 3. Solve right triangles in applied problems with trig ratios and the Pythagorean Theorem.
• MA.HS.G-SRT.D Apply trigonometry to any triangle.
  • 1. (+) Derive the triangle area formula A = ½ab sin C.
  • 2. (+) Prove the Law of Sines and the Law of Cosines and apply them.
  • 3. (+) Apply the Laws of Sines and Cosines to find unknown sides and angles in any triangle.
• MA.HS.G-C.A Understand and apply circle theorems.
  • 1. Prove that all circles are similar.
  • 2. Describe relationships among inscribed, central and circumscribed angles, radii and chords, including tangent and radius.
  • 3. Construct a triangle's inscribed and circumscribed circles and prove angle facts for cyclic quadrilaterals.
  • 4. (+) Construct a tangent from an outside point to a circle.
• MA.HS.G-C.B Find arc lengths and sector areas.
  • 1. (+) Use similarity to show arc length is proportional to radius, define radians and derive the sector area formula. (being checked)
• MA.HS.G-GPE.A Move between geometric descriptions and equations of conic sections.
  • 1. Derive the equation of a circle from the Pythagorean Theorem and complete the square to find center and radius.
  • 2. (+) Derive the equation of a parabola from a focus and directrix. (being checked)
  • 3. (+) Derive equations of ellipses and hyperbolas from their foci.
• MA.HS.G-GPE.B Use coordinates to prove simple theorems algebraically.
  • 1. Prove simple geometry theorems algebraically with coordinates.
  • 2. Prove the slope tests for parallel and perpendicular lines and apply them to problems.
  • 3. Find the point that divides a directed segment in a given ratio.
  • 4. Use coordinates (e.g. the distance formula) to find polygon perimeters and triangle and rectangle areas.
• MA.HS.G-GMD.A Explain volume formulas and use them.
  • 1. Give informal arguments for circle circumference and area and for cylinder, pyramid and cone volume formulas.
  • 2. (+) Use Cavalieri's principle informally to justify the sphere volume formula and others.
  • 3. Solve problems using the formulas for the volume of a cylinder, pyramid, cone or sphere.
• MA.HS.G-GMD.B Visualize links between 2D and 3D objects.
  • 1. Identify 2-D cross-sections of 3-D objects and 3-D objects formed by rotating 2-D shapes.
• MA.HS.G-MG.A Apply geometry to modeling situations.
  • 1. Use shapes and their measures to describe real objects (e.g. a tree trunk as a cylinder).
  • 2. Use density by area or volume in modeling (e.g. people per square mile).
  • 3. Use geometric methods to solve design problems with physical limits or cost goals.