NY Math Standards (Geometry)

Prove that all circles are similar.

Identify, describe and apply relationships between the angles and their intercepted arcs of a circle.

Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle. Note: These relationships that pertain to the circle may be utilized to prove other relationships in geometric figures, e.g., the opposite angles in any quadrilateral inscribed in a circle are supplements of each other. Also includes algebraic problems built upon these concepts.

Using proportionality, find one of the following given two others: the central angle, arc length, radius or area of sector. Note: Angle measure is in degrees.

Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane.

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not. Note: Instructional strategies may include drawing tools, graph paper, transparencies and software programs.

Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that carry the polygon onto itself. Note: The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.”

Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments. Notes: Includes point reflections. A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector. A rotation requires knowing the center (point) and the measure/direction of the angle of rotation. A line reflection requires a line and the knowledge of perpendicular bisectors.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another. Notes: Instructional strategies may include graph paper, tracing paper, and geometry software. Includes point reflections. Singular transformations that are equivalent to a sequence of transformations may be utilized, such as a glide reflection. However, glide reflections are not an expectation of the course.

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.

Use 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.

Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL (Hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions.

Prove and apply theorems about lines and angles. Note: Include multi-step proofs and algebraic problems built upon these concepts. Examples of theorems include but are not limited to: Vertical angles are congruent. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. The points on a perpendicular bisector are equidistant from the endpoints of the line segment.

Prove and apply theorems about triangles. Note: Include multi-step proofs and algebraic problems built upon these concepts. Examples of theorems include but are not limited to: Angle Relationships – The sum of the interior angles of a triangle is 180 degrees; the measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle. Side Relationships – The length of one side of a triangle is less than the sum of the lengths of the other two sides; in a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length. Isosceles Triangles – Base angles of an isosceles triangle are congruent.

Prove and apply theorems about parallelograms. Notes: Include multi-step proofs and algebraic problems built upon these concepts. The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.” Examples of theorems include but are not limited to: A diagonal divides a parallelogram into two congruent triangles. Opposite sides/angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle. Additional theorems covered allow for proving that a given quadrilateral is a particular parallelogram (rhombus, rectangle, square) based on given properties.

Make, justify and apply formal geometric constructions. Notes: Examples of constructions include but are not limited to: Copy segments and angles. Bisect segments and angles. Construct perpendicular lines including through a point on or off a given line. Construct a line parallel to a given line through a point not on the line. Construct a triangle with given lengths. Construct points of concurrency of a triangle (centroid, circumcenter, incenter, and orthocenter). Construct the inscribed circle of a triangle. Construct the circumscribed circle of a triangle. Constructions of transformations. This standard is a fluency recommendation for Geometry. Fluency with the use of construction tools, physical and computational, helps students draft a model of a geometric phenomenon and can lead to conjectures and proofs.

Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Find the center and radius of a circle, given the equation of the circle. Notes: Finding the center and radius may involve completing the square. The completing the square expectation for Geometry follows Algebra I: leading coefficients will be 1 (after possible removal of GCF) and the coefficients of the linear terms will be even. Completing the square may yield a fractional radius.

Graph circles given their equation. Note: For circles being graphed, the center will be an ordered pair of integers and the radius a positive integer.

On the coordinate plane, algebraically prove geometric theorems and properties. Notes: Examples include but not limited to: Given points and/or characteristics, prove or disprove a polygon is a specified quadrilateral or triangle based on its properties. Given a point that lies on a circle with a given center, prove or disprove that a specified point lies on the same circle. This standard is a fluency recommendation for Geometry.

On the coordinate plane, explore the proof for the relationship between slopes of parallel and perpendicular lines. Note: This standard is a fluency recommendation for Geometry.

On the coordinate plane, determine if lines are parallel, perpendicular, or neither, based on their slopes.

On the coordinate plane, apply properties of parallel and perpendicular lines to solve geometric problems.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Note: Midpoint formula is a derivative of this standard.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. ★ Note: This standard is a fluency recommendation for Geometry.

Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★

Identify the shapes of plane sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Note: Plane sections are not limited to being parallel or perpendicular to the base.

Use geometric shapes, their measures, and their properties to describe objects. ★

Apply concepts of density based on area and volume of geometric figures in modeling situations. ★

Apply geometric methods to solve design problems. ★ Note: Applications may include designing an object or structure to satisfy constraints such as area, volume, mass and cost.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Notes: The center and scale factor of the dilation must always be specified with dilation. A translation displaces every point in the plane by the same distance (in the same direction) and can be described using a vector. A rotation requires knowing the center (point) and the measure/direction of the angle of rotation. A line reflection requires a line and the knowledge of perpendicular bisectors.

Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar.

Prove and apply similarity theorems about triangles. Notes: Include multi-step proofs and algebraic problems built upon these concepts. Examples of theorems include but are not limited to: If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally (and conversely). The length of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse. The centroid of the triangle divides each median in the ratio 2:1.

Use congruence and similarity criteria for triangles to solve problems algebraically and geometrically. Notes: ASA, SAS, SSS, AAS, and Hypotenuse-Leg (HL) theorems are valid criteria for triangle congruence. AA~, SAS~, and SSS~ are valid criteria for triangle similarity. This standard is a fluency recommendation for Geometry.

Use congruence and similarity criteria for triangles to prove relationships in geometric figures. Notes: ASA, SAS, SSS, AAS, and Hypotenuse-Leg (HL) theorems are valid criteria for triangle congruence. AA~, SAS~, and SSS~ are valid criteria for triangle similarity. This standard is a fluency recommendation for Geometry.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems. ★ Note: Special right triangles refer to the 30-60-90 and 45-45-90 triangles.

Justify and apply the formula A = (1/2)ab sin(C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.