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Unit 1 -Tools of Geometry Definitions Point: A location in space Line: A series of points that extends in 2 opposite directions Plane: A flat Surface that has no thickness Collinear Points: Points that are on the same line Coplanar: Points and lines on the same plane Line Segment: A part of a line consisting of two endpoints and all the points between them Ray: A part of a line consisting of one endpoint and all the points on one side Parallel Lines: Lines in the same plane that do NOT intersect Skew Lines: Lines that are NON COPLANAR and do not intersect Angle Addition Postulate Segment Addition Postulate Mid Point Postulate Vertical Angles: Complimentary angles: angles that Add up to 90 degrees. (angles that form right angles) Supplementary Angles: Angles that Add up to 180 degrees. (Angles that are on a line) Adjacent Angles: 2 coplanar angles that share a common side and vertex 5 fundamental Loci Locus: A set of points that satisfy a given condition 1) A locus of points equidistant from a given point = a circle Ex: A locus of points 2 units from a point P 2) A locus of points equidistant from a line segment = a track Ex: The locus of points 5 units from segment 3) A locus of points equidistant from a line = 2 lines Ex: 3 units from line m 4) A locus of points equidistant from 2 lines(parallel and intersecting) = one line(parallel) or intersecting lines (intersecting) Ex: Equidistant from lines l and m(parallel) 5) A locus of points equidistant from 2 points = one line between them Ex: equidistant from points A and B l and m (intersecting) Unit 2 – Reasoning and Proof Definitions Conditional: “if…-then…” statement Hypothesis: “if” part of the statement, Conclusion – “then” part of the statement Converse: Switches the hypothesis and conclusion Inverse: Negates the hypothesis and conclusion Contrapositive: both switches and negates the hypothesis and conclusion (This is logically equivalent to the original conditional statement) Biconditional: “if and only if” statement Ex: If you are reading this, then you are studying Converse: If you are studying, then you are reading this Inverse: If you are not reading this, then you are not studying Contrapositive: If you are not studying, you are not reading this Law of Detachment: If the conditional statement is true, and the hypothesis is true, then the statement is true. Syllogism Properties of Equality Addition property: If a=b then a+c = b+c Subtraction Property: If a=b then a-c = b-c Multiplication Property: If a=b then ac=bc Division Property: If a=b then a/c=b/c Reflexive Property: a=a Symmetric Property: If a=b, then b=a Transitive Property: If a=b and b=c, then a=c Substitution property: If a=b, then “b” can replace “a” Distributive Property: a(b+c) = ab+bc Properties of Congruence Reflexive Property: Symmetric Property: Transitive Property: Unit 3 – Parallel and Perpendicular Lines Transversal: Line t Alternate Interior Angles: Angles are equal Same Side Interior Angles: Angles add to 1800 Corresponding Angles: Angles are equal Converse Statements: If the above angles are true, then you have line l parallel to line m. Triangles and Polygons Types of Triangles Equiangular/Equilateral: All sides and angles are congruent Acute: All angles are less than 900 Right: A triangle with one 900 angle Obtuse: A triangle with one angle that is bigger than 900 Isosceles: A triangle with 2 equal sides (and two equal angles) Scalene: A triangle with no equal sides Triangle Sum Theorem: The sum of the measures of the angles in a triangle is 1800. Exterior Angle Theorem: Polygon Interior Angles Theorem: The sum of the measures of the angles inside a polygon is given as (n-2)180 where “n” is the number of sides of the polygon Polygon Exterior Angles Theorem: The sum of the degrees in the exterior angles of a polygon is equal to 3600 Regular Polygon: All sides and angles are congruent Lines/Slopes Slope intercept form of a line: y = mx+b ; m = slope and b = y-intercept Standard form of a line: Ax + By = C Parallel Lines: Lines that have the SAME slope Perpendicular lines: Lines whose slopes are negative reciprocals Unit 4 – Congruent Triangles Properties of Congruence Reflexive Property: Symmetric Property: Transitive Property Third Angle Theorem: If two angles in a Triangle are congruent, then the third angle is also congruent SSS – Two triangles are congruent if their sides are congruent SAS – Two Triangles are congruent is two consecutive sides and the angle between them are congruent ASA – Two triangles are congruent if two consecutive angles and the side between them are congruent AAS – Two triangles are congruent if two consecutive angles and the third side are congruent CPCTC – Corresponding parts of congruent triangles are congruent HL – In a right triangle if the hypotenuse and legs are congruent, then the triangles are congruent Isosceles Isosceles Triangle theorem: If two sides of a triangle are congruent, the base angles are congruent. AND If the base angles are congruent the sides are congruent. Unit 5 –Relationships within Triangles Midsegment: A midsegment is a segment connecting the midpoints of two of the sides Theorem: The midsegment is half the length of the third side and parallel to it. Bisector: A line that cuts the segment (or angle) into two equal parts. Concurrent: Three or more lines that intersect at the same point Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle Incenter: The point of Concurrency of the angle bisectors of a triangle Medians: The medians of a triangle are concurrent at a point that is 2/3 the distance from the vertex to the midpoint on the opposite side. The median is drawn by drawing a line from the vertex of an angle to the midpoint on the other side. Centriod: The point of concurrency connecting the medians of a triangle Altitudes: A line segment connecting a point on the vertex angle to the side at a right angle Orthocenter: The point of concurrency of the medians of a triangle Triangle Properties: 1) The largest angle is opposite the longest side, the smallest angle is opposite the shortest side. 2) The sum of any two sides must be greater than the third side Unit 6 - Quadrilaterals Special Quadrilaterals Parallelogram: Opposite sides are parallel and congruent Rhombus: Four congruent sides Rectangle: Four right angles Square: Four right angles and four equal sides Kite: Two pairs of adjacent sides congruent, and no opposite sides congruent Trapezoid: Only one pair of parallel sides Parallelogram properties(and methods of proof) 1) Opposite sides are congruent 2) Opposite sides are parallel 3) Opposite angles are congruent 4) Diagonals Bisect each other P1) If both pairs of sides are congruent, then it is a parallelogram P2) If both pairs of opposite angles are congruent, then it is a parallelogram P3) If the diagonals of a parallelogram bisect each other, then it is a parallelogram P4) If one pair of opposite sides are BOTH parallel and congruent, then it is a parallelogram Special Parallelogram Properties(and methods of proof) 1) Each diagonal of a rhombus bisects the angles of a rhombus 2) The diagonals of a Rhombus are perpendicular 3) The diagonals of a rectangle are congruent P1) If one diagonal of a parallelogram bisects two angles, then it is a rhombus P2) If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus P3) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle Trapezoids and Kites(and methods of proof) 1) The base angles of an Isosceles trapezoid are congruent 2) The diagonals of an Isosceles triangle are congruent 3) The Diagonals of a Kite are perpendicular Trapezoid Midsegment Theorem Unit 7 – Similar Triangles Similarity – Polygons are similar if there angles are congruent corresponding sides are proportional AA – Two Triangles are similar if two angles from one triangle are congruent to two angles in the other triangle. SAS Similarity – Two triangles are similar is two sides are in proportion and the angle between is congruent. SSS Similarity – If three sides are in proportion then the triangles are similar Geometric Mean – Side Splitter Theorem – If a line is parallel to one side of a triangle and intersects two other sides, then it divides those sides proportionally. Pythagorean Theorem – a2 + b2 = c2 in a right triangle where “c” is the hypotenuse. And the it is true…If a2+b2 = c2 then it is a right triangle. Common Pythagorean Triples: 3,4,5 5,12,13 8,15,17 7,24,25 Special Right Triangles – Sin(x) = Opposite Hypotenuse Cos(x) = Adjacent Hypotenuse Tan(x) = Opposite Adjacent Unit 8 - Circles Standard form of a Circle (x-a)2 + (y-b)2 = r2 Arc Properties – Area of a Circle = 2 Tangent Lines to a Circle – A tangent line is always perpendicular to the radius of the circle. Chord Properties – 1) Congruent central angles have congruent chords 2) Congruent chords have congruent arcs 3) Congruent Arcs have congruent central angles 4) Chords equidistant from the center are congruent 5) A diameter that is perpendicular to the chord, bisects the chord(diagram) Unit 9 - Transformations Transformation – A geometric Figure that can change size, shape or position Preimage – The original figure Image – the resulting figure after a transformation Isometry – a transformation that keeps the figures congruent(Only a dilation will change Isometry) Composition – A combination of two or more transformations Reflection – a flip (opposite orientations) Reflection over the x-axis (a,b) = (a,-b) Reflection over the y-axis (a,b) = (-a,b) Rotation – a turn 900 around the origin (a,b) = (-b,a) 1800 around the origin (a,b) = (-a,-b) Dilation – increase or decrease in size of figure Translation – A movement of a figure Symmetry – an Isometry that maps a figure onto itself(line, rotational, point) Unit 10 - Solids Lateral Area = Sum of the areas of the lateral faces (do not include the bases) Surface area = sum of the areas of the faces (include the bases) Lateral area of a cone = Volume of a Prism = Area of the base x height Volume of a pyramid = 1/3 x area of the base x height 2 Surface area of a sphere = Volume of a sphere = 4/3 2 Area and volumes of Similar solids: If two figures are in the ratio of A:B The areas are in the ration of A2:B2 The volumes are in the ration of A3:B3

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