High School Geometry Curriculum

Below are skills needed, with links to resources to help with that skill. We also encourage plenty of exercises and book work. Curriculum Home

Important: this is a guide only.
Check with your local education authority to find out their requirements.

High School Geometry | Measurement
☐ Convert between radian and degree measures
☐ Define a Steradian and know its relationship to square degrees.
High School Geometry | Geometry (Plane)
☐ Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
☐ Determine the length of an arc of a circle, given its radius and the measure of its central angle
☐ Construct a bisector of a given angle, using a straightedge and compass, and justify the construction
☐ Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction
☐ Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction
☐ Construct an equilateral triangle, using a straightedge and compass, and justify the construction
☐ Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles
☐ Solve problems using compound loci
☐ Identify corresponding parts of congruent triangles and other figures
☐ Investigate, justify, and apply the isosceles triangle theorem and its converse
☐ Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem
☐ Based on the measure of given angles formed by the transversal and the lines, determine whether two or more lines cut by a transversal are parallel.
☐ Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons
☐ Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons
☐ Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals
☐ Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares, kites) involving their angles, sides, and diagonals
Kite
☐ Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals
☐ Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids
Kite
☐ Investigate, justify, and apply theorems about similar triangles
☐ Given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle, investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle.
☐ Investigate, justify, and apply theorems about mean proportionality: * the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse * the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg
☐ Investigate, justify, and apply theorems regarding chords of a circle: * perpendicular bisectors of chords * the relative lengths of chords as compared to their distance from the center of the circle
☐ Investigate, justify, and apply theorems about tangent lines to a circle: * a perpendicular to the tangent at the point of tangency * two tangents to a circle from the same external point * common tangents of two non-intersecting or tangent circles
☐ Investigate, justify, and apply theorems about two lines intersecting a circle when the vertex is inside the circle (two chords) or on the circle (tangent and chord).
☐ Investigate, justify, and apply theorems regarding segments intersected by a circle: * along two tangents from the same external point * along two secants from the same external point * along a tangent and a secant from the same external point * along two intersecting chords of a given circle
☐ Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
☐ Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections
☐ Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)
☐ Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)
☐ Investigate, justify, and apply the properties that remain invariant under similarities
☐ Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism
☐ Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x=0, y=0, and y=x, and dilations centered at the origin
☐ Construct the center of a circle using a straight edge and compass.
☐ Calculate the area of a segment of a circle, given the measure of a central angle and the radius of the circle
☐ Construct a circle touching three points using a straight edge and compass.
☐ Circumscribe a circle on a triangle using a straight edge and compass.
☐ Construct a triangle with three known sides using a ruler and compass, and justify the construction
☐ Cut a line into n equal segments using a straightedge and compass, and justify the construction
☐ Construct a circle inscribed within a triangle (incircle) using a ruler and compass, and justify the construction.
☐ Construct a pentagon using a ruler and compass, and justify the construction.
☐ Construct a tangent from a point to a circle using a ruler and compass, and justify the construction.
☐ Know that the apothem of a regular polygon is the radius of its incircle, and know its relationship to the radius of the circumcircle of the polygon or the length of side of the polygon.
☐ Calculation of the area of a regular polygon from the number of sides and either the length of side, radius of the circumcircle or length of apothem.
☐ Investigate, justify, and apply theorems about the number of diagonals of regular polygons.
☐ Investigate the properties of the pentagram, and its relationship to the golden ratio.
☐ Use a ruler and drafting triangle to construct a line parallel to a given line and passing through a given point, or to construct a line perpendicular to a given line at a given point.
☐ Understand that a plane is a flat surface with no thickness that goes on forever.
☐ Know how to find the ratio of the areas of similar shapes given the ratio of their lengths.
☐ Investigate and understand circle theorems including the Angle at the Center Theorem, the Angles Subtended by Same Arc Theorem and The Angle in the Semicircle Theorem.
☐ Investigate cyclic quadrilaterals and know that opposite angles of a cyclic quadrilateral are supplementary.
☐ Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle using a straightedge and compass, and justify the constructions.
☐ Prove that all circles are similar.
☐ Calculate unknown lengths inside a circle using the Intersecting Chords Theorem.
☐ Calculate unknown lengths outside a circle using the Intersecting Secants Theorem.
☐ Investigate, justify, and apply theorems about two lines intersecting a circle when the vertex is outside the circle (two tangents, two secants, or tangent and secant).
High School Geometry | Geometry (Solid)
☐ Use formulas to calculate volume and surface area of rectangular solids and cylinders
☐ Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them
☐ Know and apply that the lateral edges of a prism are congruent and parallel
☐ Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal
☐ Know and apply that the volume of a prism is the product of the area of the base and the altitude
☐ Apply the properties of a regular pyramid, including: # lateral edges are congruent # lateral faces are congruent isosceles triangles # volume of a pyramid equals one-third the product of the area of the base and the altitude
☐ Apply the properties of a cylinder, including: * bases are congruent * volume equals the product of the area of the base and the altitude * lateral area of a right circular cylinder equals the * product of an altitude and the circumference of the base
☐ Apply the properties of a right circular cone, including: * lateral area equals one-half the product of the slant height and the circumference of its base * volume is one-third the product of the area of its base and its altitude
☐ Apply the properties of a sphere, including: * the intersection of a plane and a sphere is a circle * a great circle is the largest circle that can be drawn on a sphere * two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles * surface area is 4 pi r^2 * volume is (4/3) pi r^3
☐ Know and apply that through a given point there passes one and only one plane perpendicular to a given line
☐ Know and apply that through a given point there passes one and only one line perpendicular to a given plane
☐ Know and apply that two lines perpendicular to the same plane are coplanar
☐ Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane
☐ Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane
☐ Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane
☐ Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines
☐ Know and apply that if two planes are perpendicular to the same line, they are parallel
☐ Understand what is meant by the cross section of a prism, cylinder, pyramid, sphere or torus and recognize the shape of the cross section.
☐ Understand what is meant by the dihedral angle between two planes.
☐ Understand Euler's Formula connecting the numbers of faces, vertices and edges of the Platonic solids and many other solids.
☐ Understand why there are exactly five Platonic solids.
☐ Know the properties of a torus, including the formulas for surface area and volume.
☐ Use formulas to calculate the surface areas and volumes of the tetrahedron, the cube, the octahedron, the dodecahdron and the icosahedron.
☐ Compare the volumes and surface areas of a cone (radius r, height 2r), a sphere (radius r) and a cylinder (radius r, height 2r).
☐ Calculate the cross-sectional area of a partially filled horizontal cylinder using the formula Area = cos-1((r - h)/r)r^2 - (r - h)sqrt(2rh - h^2) and hence calculate its volume.
High School Geometry | Coordinates
☐ Understand Polar Coordinates, and how to convert from Cartesian coordinates to polar coordinates and vice versa.
High School Geometry | Trigonometry
☐ Find the sine, cosine, and tangent ratios (or their reciprocals) of an angle of a right triangle, given the lengths of the sides
☐ Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle
☐ Find the measure of a side of a right triangle, given an acute angle and the length of another side
☐ Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides
☐ Express and apply the six trigonometric functions as ratios of the sides of a right triangle, and know the trigonometric identities: tan(x) = sin(x)/cos(x) etc
☐ Know the exact and approximate values of the sine, cosine, and tangent of 0, 30, 45, 60, 90, 180, and 270 degree angles
☐ Sketch and use the reference angle for angles in standard position
☐ Know and apply the co-function and reciprocal relationships between trigonometric ratios
☐ Use the reciprocal and co-function relationships to find the values of the secant, cosecant, and cotangent of 0, 30, 45, 60, 90, 180, and 270 degree angles
☐ Sketch the unit circle and represent angles in standard position
☐ Find the value of trigonometric functions, if given a point on the terminal side of angle (theta)
☐ Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function
☐ Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent
☐ Sketch the graphs of the inverses of the sine, cosine, and tangent functions
☐ Determine the trigonometric functions of any angle, using technology
☐ Justify the Pythagorean identities
☐ Solve simple trigonometric equations for all values of the variable from 0 degrees to 360 degrees (four quadrants)
☐ Determine amplitude, period, frequency, phase shift and vertical shift given the graph or equation of a periodic function
☐ Sketch and recognize one cycle of a function of the form y = A sin(Bx) or y = A cos(Bx)
☐ Sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x)
☐ Write the trigonometric function that is represented by a given periodic graph
☐ Solve for an unknown side or angle, using the Law of Sines
☐ Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle
☐ Determine the solution(s) of triangles from the SSA situation (ambiguous case)
☐ Apply the angle sum and difference formulas for trigonometric functions
☐ Apply the double-angle and half-angle formulas for trigonometric functions
☐ Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles
☐ Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle
☐ Investigate, justify, and apply the triangle inequality theorem
☐ Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle
☐ Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1
☐ Establish similarity of triangles, using the following theorems: AA, SAS, and SSS
☐ Investigate, justify, apply, and prove the Pythagorean theorem and its converse. Include proof of the Pythagorean Theorem using triangle similarity.
☐ Sketch and recognize the graphs of the functions y=sin(x), y=cos(x) and y=tan(x)
☐ Find the area of a triangle given the lengths of its three sides, using Heron's formula.
☐ Recognize that an AAA triangle is impossible to solve.
☐ Use the symmetric properties of an equilateral triangle to solve triangles by reflection.
☐ Be familiar with the triangle identities that are true for all triangles: The Law of Sines, The Law of Cosines and the Law of Tangents.
☐ Know and apply the opposite angle identities: sin(-A) = -sin(A), cos(-A) = cos(A) and tan(-A) = -tan(A)
☐ Know how to find the values of sine, cosine and tangent in each of the four quadrants; including determining the correct sign.
☐ Solve for an unknown side or angle, using the Law of Cosines
☐ Solve a triangle using the Law of Sines and the Law of Cosines
☐ Use the magic hexagon to remember trigonometric identities
☐ Use the Pythagorean Theorem in three dimensions, including calculating the length of a space diagonal of a cuboid given the length, width and height.
☐ Know how to express a bearing using three-figure bearings, and how to convert between three-figure bearings and the principal compass bearings.