# 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

☐ Define radian measure

☐ Radians

☐ Convert between radian and degree measures

☐ Radians

☐ 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).

☐ Polygons

☐ Circle

☐ Determine the length of an arc of a circle, given its radius and the measure of its central angle

☐ Radians

☐ Construct a bisector of a given angle, using a straightedge and compass, and justify the construction

☐ Bisect

☐ Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction

☐ Bisect

☐ 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

☐ Bisect

☐ 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

☐ Polygons

☐ Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons

☐ 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

☐ Rhombus

☐ 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

☐ Rhombus

☐ 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

☐ Circle

☐ Bisect

☐ 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)

☐ Similar

☐ Investigate, justify, and apply the properties that remain invariant under similarities

☐ Similar

☐ Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism

☐ Similar

☐ 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.

☐ Ratios

☐ Similar

☐ 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.

☐ Circle

☐ 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.

☐ Circle

☐ 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

☐ Pyramids

☐ 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

☐ Sphere

☐ 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.

☐ Pyramids

☐ Torus

☐ Sphere

☐ 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.

☐ Torus

☐ 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).

☐ Sphere

☐ 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

☐ Centroid

☐ 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.