M.GHS: High School Geometry

High School Geometry

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High School Geometry develops geometric reasoning through proof, transformation, and coordinate methods. Congruence and similarity are established via rigid motions and dilations, and students prove triangle and parallelogram theorems using these foundations. Circles, coordinate geometry, geometric constructions, right triangle trigonometry, and modeling with geometric shapes and volume formulas complete the course.

M.GHS.1Know 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.M.GHS.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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.M.GHS.11Prove 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.M.GHS.12Make 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.M.GHS.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.M.GHS.14Verify experimentally the properties of dilations given by a center and a scale factor.M.GHS.15Given 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.M.GHS.16Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.M.GHS.17Prove 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.M.GHS.18Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.M.GHS.19Understand 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.M.GHS.2Represent transformations in the plane using, for example, 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).M.GHS.20Explain and use the relationship between the sine and cosine of complementary angles.M.GHS.21Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.M.GHS.22Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.M.GHS.23Prove the Laws of Sines and Cosines and use them to solve problems.M.GHS.24Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.M.GHS.25Give 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.M.GHS.26Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.M.GHS.27Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.M.GHS.28Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).M.GHS.29Use coordinates to prove simple geometric theorems algebraically. (e.g., 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).M.GHS.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.M.GHS.30Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.)M.GHS.31Find the point on a directed line segment between two given points that partitions the segment in a given ratio.M.GHS.32Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem.M.GHS.33Derive the equation of a parabola given a focus and directrix.M.GHS.34Prove that all circles are similar.M.GHS.35Identify 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.M.GHS.36Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.M.GHS.37Construct a tangent line from a point outside a given circle to the circle.M.GHS.38Derive 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.M.GHS.39Derive 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.M.GHS.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.M.GHS.40Use coordinates to prove simple geometric theorems algebraically. (e.g., 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).)M.GHS.41Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).M.GHS.42Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).M.GHS.43Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.M.GHS.44Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that 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.M.GHS.45Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.M.GHS.46Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.M.GHS.47Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.M.GHS.48Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.M.GHS.49Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.M.GHS.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.M.GHS.50Use permutations and combinations to compute probabilities of compound events and solve problems.M.GHS.51Use probabilities to make fair decisions (e.g., drawing by lots and/or using a random number generator).M.GHS.52Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).M.GHS.53Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).M.GHS.54Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).M.GHS.55Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).M.GHS.6Use 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.M.GHS.7Use 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.M.GHS.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.M.GHS.9Prove 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.M.GHS.APApplications of ProbabilityM.GHS.CCircles With and Without CoordinatesM.GHS.CAConnecting Algebra and Geometry Through CoordinatesM.GHS.CPCCongruence, Proof, and ConstructionsM.GHS.E3DExtending to Three DimensionsM.GHS.MGModeling with GeometryM.GHS.SPTSimilarity, Proof, and Trigonometry
Example Problems
Point A is at and point C is at .
Find the coordinates of point B on such that .
A rectangle has four right angles.
A parallelogram has two pairs of parallel sides.
Is every rectangle also a parallelogram?
Point A is at and point C is at .
Find the coordinates of point B on such that .
Polygon Y has an area of 11 square units. Celia drew a scaled version of Polygon Y using a scale factor of 3 and labeled it Polygon Z.
What is the area of Polygon Z?
A square tile measures 15 centimeters on each side.

What is the diagonal distance across the tile?
Write your answer as a decimal rounded to the nearest hundredth.
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