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Vocabulary Construct Congruent Angle Arc Circle Point Compass Distance Perpendicular Parallel Transversal Triangle Equilateral Equilateral triangle Radius Point of concurrency Median Altitude Perpendicular bisector Line segment Concepts/Topics NY State Performance Indicators Skills to Review Project/Lab UNIT 1: CONSTRUCTION 9/13 – CHAPTER 1 G.G.17 Construct a bisector of a given Compasses help us create congruent arcs or angle, using a straightedge and compass, circles that intersect at points that allow us and justify the construction to bisect angles. G.G.19 Construct lines parallel (or A compass allows us to replicate the perpendicular) to a given line through a distance of two points given point, using a straightedge and Using arcs or circles greater than ½ the given distance helps identify perpendicular compass, and justify the construction G.G.18 Construct the perpendicular lines bisector of a given segment, using a If 2 lines cut by a transversal make congruent angles, then the lines are parallel straightedge and compass, and justify the construction (the goal of the construction of a parallel G.G.20 Construct an equilateral triangle, line is to construct a pair of congruent using a straightedge and compass, and angles) justify the construction Equilateral triangles have equal sides. G.G.21 Investigate and apply the These sides will be radii of congruent concurrence of medians, altitudes, angle circles that can be constructed using the bisectors, and perpendicular bisectors of compass. triangles Triangles can have up to 4 points of concurrency Perpendicular bisectors of a triangle share the properties of both medians and altitudes UNIT 2: REASONING AND PROOF CHAPTER 2 G.G.24 Determine the negation of a Proof is a justification that is logically valid statement and establish its truth value and based on definitions, postulates, and G.G.25 Know and apply the conditions theorems. under which a compound statement Logical arguments consist of a set of (conjunction, disjunction, conditional, premises or hypotheses and a conclusion. 1 Parallel Perpendicular Slope Line Line segment Point Equation Midpoint Bisect Bisector Length Distance Endpoint Perpendicular bisector Distance formula Midpoint formula biconditional) is true Reasoning and proofs are fundamental to mathematics and help us prove or disprove G.G.26 Identify and write the inverse, converse, and contrapositive of a given various conjectures. conditional statement and note the Only 1 counterexample is necessary to prove a statement false, but 1 example does logical equivalences G.G.27 Write a proof arguing from a given not prove the truth value of a statement. hypothesis to a given conclusion Inverse, converse, and contrapositive statements relate to the original conditional statement and vary in whether they are always logically valid. UNIT 3: PARALLEL AND PERPENDICULAR LINES CHAPTER 3 Introduce formal proofs here G.G.35 Determine if two lines cut by a transversal are parallel, based on the Lines are parallel if alternate interior and measure of given pairs of angles formed exterior angles are congruent by the transversal and the lines Alternate interior and exterior angles are G.G.62 Find the slope of a perpendicular congruent if lines are parallel (this is the line, given the equation of a line converse of statement 1) G.G.63 Determine whether two lines are There is a special relationship between the parallel, perpendicular, or neither, given slopes of 2 parallel lines and the slopes of 2 their equations perpendicular lines. G.G.64 Find the equation of a line, given Slopes of parallel lines are equal a point on the line and the equation of a Slopes of perpendicular lines are opposite line perpendicular to the given line reciprocals of one another We can use the relationship of the slopes of G.G.65 Find the equation of a line, given a point on the line and the equation of a parallel and perpendicular lines to prove line parallel to the desired line whether two equations produce parallel or G.G.66 Find the midpoint of a line perpendicular lines segment, given its endpoints The relationship of the slopes of parallel and perpendicular lines allows one to graph G.G.67 Find the length of a line segment, given its endpoints and find the equations of lines G.G.68 Find the equation of a line that is The mid-points formula finds the halfway the perpendicular bisector of a line point for the x- and y-coordinates segment, given the endpoints of the line The distance formula is derived from the segment Pythagorean theorem (the distance is the hypotenuse of the right triangle formed by Write a proof arguing from a given hypo Slope Finding the equation of a line given two points Finding the equation of a line given slope and one point Pythagorean Theorem 2 the x- and y-coordinates) We can use the midpoint and our understanding of perpendicular lines to find the equation of a perpendicular bisector within a coordinate plane UNIT 4: TRIANGLE PROPERTIES AND CONGRUENCE CHAPTERS 4 & 5 What does it mean for triangles to be congruent? G.28 Determine the congruence of two triangles using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL) given How can I prove that two triangles are sufficient information about the sides congruent? and/or angles of two congruent triangles G.29 Identify corresponding parts of What are some of the defining characteristics congruent triangles of a triangle? G.30 Investigate, justify, and apply theorems about the sum of the measures What characterizes different types of of the angles of a triangle triangles? G.32 Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem G.31 Investigate, justify and apply the isosceles triangle theorem and its converse G.33 Investigate, justify and apply the triangle inequality theorem G.34 Determine the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle UNIT 5: TRIANGLE SIMILARITY CHAPTER 8 3 Alternate interior angles Alternate exterior angles Corresponding angles Vertical angles Supplementary angles Linear pairs Parallelogram Diagonal Opposite angles Rhombus Square G.44 Establish similarity of triangles, using the following theorems: AA, SAS, and SSS G.45 Investigate, justify, and apply theorems about similar triangles G.46 Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle G.42 Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle G.47 Investigate, justify, and apply theorems about mean proportionality UNIT 6: QUADRILATERALS CHAPTER 6 G.G.38 Investigate, justify, and apply Parallelograms: theorems about parallelograms involving have 2 sets of parallel sides their angles, sides, and diagonals diagonals bisect one another G.G.39 Investigate, justify, and apply have opposite angles that are always theorems about special parallelograms congruent (rectangles, rhombuses, squares) Rhombuses and squares have congruent involving their angles, sides, and sides diagonals Rectangles and squares have 4 right angles G.G.40 Investigate, justify, and apply Quadrilateral is the broad category that includes more specific 4-sided polygons like theorems about trapezoids (including isosceles trapezoids) involving their parallelograms, rhombuses, rectangles, angles, sides, medians, and diagonals squares, and trapezoids G.G.41 Justify that some quadrilaterals The sum of a polygons’ interior angles is are parallelograms, rhombuses, determined by 180(n-2) rectangles, squares, or trapezoids A polygon’s exterior angles can have a 4 Rectangle Right angle Quadrilateral Polygon Trapezoid Interior angle Exterior angle negative or a positive measurement as they are always measured counterclockwise. Every pair of interior and exterior angles are supplementary (add up to 180 degrees) G.G.36 Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons G.G.37 Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons UNIT 7: TRANSFORMATIONS G.54 Define, investigate, justify, and apply An isometry of the coordinate plane is a isometries in the plane (rotations, linear transformation which preserves reflections, translations, glide reflections) length. Put another way, isometries Note: Use proper function notation. produce congruent figures. G.55 Investigate, justify, and apply the We can use our existing knowledge of properties that remain invariant under perpendicularity, parallelism, and translations, rotations, reflections, and congruence to make generalizations about glide reflections the results of various transformations. G.56 Identify specific isometries by Not all transformations are isometries. observing orientation, numbers of Dilations produce figures that are similar, invariant points, and/or parallelism but not congruent. G.57 Justify geometric relationships It is possible to perform more than one transformation on the same figure to obtain (perpendicularity, parallelism, congruence) using transformational the image from the preimage techniques (translations, rotations, Reflections result in symmetry about a reflections) point or line. G.58 Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries) G.59 Investigate, justify, and apply the properties that remain invariant under similarities G.60 Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism G.61 Investigate, justify, and apply the analytical representations for 5 translations, rotations about the origin of 90º and 180º, reflections over the lines x=0, y=0 , and y=x , and dilations centered at the origin UNIT 8: CIRCLES AND LOCI CHAPTER 10 G.71 Write the equation of a circle, given A circle includes all points equidistant, r, its center and radius or given the from the center (h, k). endpoints of a diameter G.72 The equation of a circle includes the center Write the equation of a circle, given its point and radius graph We can figure out the location of the center Note: The center is an ordered pair of and the length of the radius of a circle simply by looking at the equation of a circle integers and the radius is an integer. The center (h, k) has the opposite sign from G.73 Find the center and radius of a how it appears in the equation The use of square roots allows us to find the circle, given the equation of the circle in center-radius form center of the circle. G.74 Graph circles of the form Loci are sets of points that satisfy 1 given condition, usually involving a distance from another object. There are 5 main scenarios that loci problems involve. Compound loci are sets of points that satisfy more than 1 given condition. Diameter Chord Center Diameter Radius Tangent line Arc length Circumference Pi Arc Secant (x h)2 (y k)2 r2 G.22 Solve problems using compound loci G.23 Graph and solve compound loci in the coordinate plane UNIT 9: TANGENTS, SECANTS, AND CHORDS G.G.49 Investigate, justify, and apply Diameters are the longest chords in any theorems regarding chords of a circle: circle. O perpendicular bisectors of chords Chords are related uniquely to circles and O the relative lengths of chords as their distance from the center compared to their distance from the Perpendicular bisectors of chords also center of the circle bisect the circle. Put another way, if a G.G.50 Investigate, justify, and apply diameter is perpendicular to a chord, it theorems about tangent lines to a circle: bisects that chord. O a perpendicular to the tangent at the The radius is always perpendicular to the point of tangency tangent line O two tangents to a circle from the same Every point outside a circle can have 6 Ray exactly two tangent lines extended to the circle Arc lengths are related to the entire circle as a fraction of the overall circumference of the circle (n/360 * 2∏r) Chords that are parallel create congruent arcs. We can make generalizations about the length of line segments based upon the number of times they intersect a circle and whether the intersection of 2 line segments is inside or outside the circle. external point O common tangents of two nonintersecting or tangent circles G.G.51 Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is: O inside the circle (two chords) O on the circle (tangent and chord) O outside the circle (two tangents, two secants, or tangent and secant) G.G.52 Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines G.G.53 Investigate, justify, and apply theorems regarding segments intersected by a circle: O along two tangents from the same external point O along two secants from the same external point O along a tangent and a secant from the same external point O along two intersecting chords of a given circle UNIT 10: COORDINATE PROOFS Wrap up year with coordinate proofs to G.G.69 Investigate, justify, and apply review coordinate geometry, parallel and the properties of triangles and perpendicular lines, and properties of quadrilaterals in the coordinate quads and triangles plane, using the distance, midpoint, and slope formulas UNIT 11: AREA, VOLUME, AND PERIMETER CHAPTERS 11 & 12 7 3-Dimensional 2-Dimensional Volume Area Perimeter Prism Base Height Altitude Face Regular pyramid Cylinder Right circular cone Sphere Plane Line Point Intersection Perpendicular Contain Coplanar Normal G.G.10 Know and apply that the lateral Volume is a measure of 3-dimensional edges of a prism are congruent and space like the amount of liquid that a parallel container could hold. G.G.11 Know and apply that two prisms Volume of a prism is the area of the base have equal volumes if their bases have times the height (or altitude) (v = l * w * h) A pyramid has a square or rectangular base equal areas and their altitudes are equal G.G.12 Know and apply that the volume and four slanted, triangular sides. At a minimum, the opposing faces are congruent of a prism is the product of the area of the base and the altitude Lateral area is the area of the lateral faces. G.G.13 Apply the properties of a regular Surface area is lateral area plus the area of pyramid the base. G.G.14 Apply the properties of a “Height” refers to the perpendicular cylinder distance from the center of the base to the common vertex where all lateral faces meet. G.G.15 Apply the properties of a right circular cone “Slant height” refers to the distance along a G.G.16 Apply the properties of a sphere lateral face from the base to the common vertex where all lateral faces meet. Volume of a cylinder is the area of the base times the height (or altitude) (v = ∏r2 * h) A cone is made of two surface pieces—a circular base and an additional lateral surface The cross section of a sphere is a circle A sphere consists of all the points whose distance from the center is less than or equal to the radius UNIT 12: LINES AND PLANES Points, lines, segments, and planes are basic G.G.1 Know and apply that if a line is perpendicular to each of two intersecting geometric structures and allow us to make lines at their point of intersection, then conclusions about more complex 2- and 3the line is perpendicular to the plane dimensional objects determined by them A plane can be defined by 3 points, 2 G.G.2 Know and apply that through a intersecting lines or a line and point not on given point there passes one and only the line one plane perpendicular to a given line A line is defined by 2 points G.G.3 Know and apply that through a Coplanar means that the given lines or 8 points lie within the same plane. When a plane contains a given line, all lines and planes that are perpendicular to the given line are also perpendicular to the plane If a line is normal to one of two parallel plane, and that line does not lie in either plane, then that line is normal to the other of the two parallel planes given point there passes one and only one line perpendicular to a given plane G.G.4 Know and apply that two lines perpendicular to the same plane are coplanar G.G.5 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 G.G.6 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 G.G.7 Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane G.G.8 Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines G.G.9 Know and apply that if two planes are perpendicular to the same line, they are parallel UNIT 13: SYSTEMS OF EQUATIONS G.G.70 Solve systems of equations involving one linear equation and one quadratic equation graphically REGENTS REVIEW Review packet for differentiated practice Whole class review games Whole class reteach difficult topics from past semester Whole class or small group instruction on quick and easy topics Full Regents – 3+ 9 10