Chapter 1 Section 1- Points and Lines as Locations Synthetic Geometry

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1 Chapter 1 Section 1- Points and Lines as Locations Synthetic Geometry A geometry studied without the use of coordinates. Coordinate The number or numbers associated with the location of a point on a line, a plane, or in space. Coordinatized Line A line on which every point is identified with exactly one number and every number is identified with a point on the line. Distance between two points on a Coordinatized Line The distance between two points on a coordinatized line is the absolute value of the difference of their coordinated. In symbols, the distance between two points with coordinates a and b is a b. Syntheitc Geometry Description of A point is an exact location. a point Description of a line A line is a set of points extending in both directions containing the shortest path between any two points on it. Section 2- Ordered Pairs as Points Plane Coordinate Geometry The study of geometrice figures using points as ordered pairs of real numbers. Standard Form of a Line The standard form of a line is Ax + By = C Horizontal Line When A = 0 and the line is of the form By = C, then the line is horizontal Vertical Line When B = 0 and the line is of the form Ax = C, then the line is vertical

2 Oblique When neither A or B equal zero, the line is oblique. Plane Coordinate Geometry Description of A point is an ordered pair of real numbers. a point Description of a line A line is the set of ordered pairs (x,y) which satisfy the equation Ax+By = C, where A and B are not both zero. Slope- Intercept Form A equation of a line in the form of y=mx+b y-intercept The value of y when x=0 Slope The ratio of the difference between y-coordinates and the difference between x-coordinates. Section 3- Other Types of Geometry Pixels A dot on a TV, computer screen, or other monitor. Discrete Geometry The study of discrete points and lines. Discrete Geometry Description of A point in discrete geometry is a dot. a point Description of a line A line in discrete geometry is a set of dots in a row. Network A union of points and segments connecting them.

3 Arcs A path from one point of a network to another point. Graph Theory The geometry of networks. Nodes An endpoint of an arc in a network. Vertices Where two or more arcs meet at a node. Traversable A network in which all the arcs may be traced exactly once without picking up the tracing instruments. Graph Theory Description of a point A point is a node of a network. Description of a line A line is an arc connecting either two nodes or one node to itself. Section 4- Undefined Terms and First Definitions Circularity The circling back that sometimes occurs when one tries to define basic terms; returning to the word which one is trying to define. Undefined Terms A term used without a specific mathematical definition. Figure A set of points. Space The set of all points Collinear Three or more points that lie on the same line.

4 Plane Figure A set of points that are all in one plane. Coplanar Figures that lie in the same plane. One- Dimensional A space in which all points are collinear. Two- Dimensional Pertaining to figures that lie in a single plane, or to their geometry. Three- Dimensional A figure whose points do not all lie in a single plane. Section 5- Postulates for Points and Lines in Euclidean Geometry Postulates A statement assumed to be true. A statement deduced from postulates, definitions, or other previously deduced theorems. Euclidean Geometry The collection of propositions about figures which includes or from which can be deduced those given by the mathematician Euclid around 250 B.C. Point-Line- Plane-Postulate Unique Line Assumption: Through any two points there is exactly one line. If the two points are in a plane, the line containing them is in the plane. Number Line Assumption: Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1. Dimension Assumption: (1) There are at least two points in space. (2) Given a line in a plane, there is at least one point in the plane that is not on the line. (3) Given a plane in space, there is at least one point in space that is not in the plane. Line Intersection Two different lines intersect in at most one point.

5 Parallel Lines Two coplanar lines which have no points in common or are identical. Section 6- Betweenness and Distance Between A point is between two other points on the same line if its coordinate is between their coordinates. Segment The set consisting of the distinct points A and B (its endpoints) and all points between A and B. Denoted AB. Ray The ray with endpoint A and containing a second points B, denoted AB, consists of the points on AB and the set of all points for which B is between the point and A. Opposite Rays AB and AC are opposite rays if and only if A is between B and C. Distance Postulate Uniqueness Property: On a coordinatized line, there is a unique distance between two points. Distance Formula: If two points on a line have coordinates x and y, the distance between them is x y. Additive Property: If B is on AC, then AB + BC = AC. Section 7 Using a Dynamic Geometry System Triangle Inequality Postulate The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

6 Section 1- The Need for Definitions Convex Set Chapter 2 A set of points in which every segment that connects points of the set lies entirely in the set. Non-convex/ Concave Set A set in which at least one segment that connects points within the set has points that lie outside of the set. Section 2- Conditional Statements Statement A sentence that is either true or false and not both. Compound Statement A sentence that combines two or more statements with some type of connective word. Conditional/ If-Then A statement of the form If.., then. Antecedent/ Hypothesis The If clause of a conditional. Consequent/ Conclusion The then clause of a conditional. Instance of a Conditional A specific case in which the antecedent of the conditional is true and its consequent it also true. Counterexample of a Conditional A specific case of a conditional for which the antecedent is true but the consequent is false. An example which shows a conjecture to be false. Section 3- Converses Converse The conditional statement formed by switching the antecedent and consequent of a given conditional. Section 4- Good Definitions Midpoint The point on the segment equidistant from the segment s endpoints.

7 Biconditional/ if and only if A statement that includes a conditional and its converse. It may be written in the form p qandq p, p q, or p if and only if q. Circle The set of points in a plane at a certain distance (its radius) from a certain point (its center). Radius A segment connecting the center of a circle or a sphere with a point on that circle or sphere. Diameter A segment connecting two points on the circle or sphere and containing the center of the circle or sphere. Section 5- Unions and Intersections of Figures Union of two sets The set of elements, which are in at least one of the sets. Intersection of two sets The set of elements, which are in both the sets. Angle The union of two rays with the same endpoint. Empty Set / Null Set The set with no elements. Section 6- Polygons Polygon The union of three or more coplanar segments such that each segment intersects exactly two others, one at each of its endpoints. Sides The segments that make up a polygon. Consecutive Vertices In a polygon, endpoints of a side. Also called adjacent.

8 Consecutive Sides In a polygon, two sides with an endpoint in common. Also caked adjacent. Diagonal A segment connecting nonconsecutive vertices of the polygon. N-gons A polygon with n sides. Triangle A polygon with three sides. Quadrilateral A polygon with four sides. Pentagon A polygon with five sides. Hexagon A polygon with six sides. Heptagon A polygon with seven sides. Octagon A polygon with eight sides. Nonagon A polygon with nine sides. Decagon A polygon with ten sides. Dodecagon A polygon with twelve sides. Polygonal Region The union of a polygon and its interior.

9 Convex Polygon A polygon is a convex polygon if and only if it corresponding polygonal region is convex. Equilateral Triangle A triangle with all three sides of equal length. Isosceles Triangle A triangle with at least two sides of equal length. Scalene Triangle A triangle with no two sides of the same length. Section 8- Conjectures Conjecture An educated guess or opinion. Section 1- Arcs and Angles Arc Chapter 3 A part of a circle connecting two points on the circle. These two points are the arc s endpoints. Measure of an Arc The amount of a circle that is traced by that arc. Semicircle An arc of a circle whose endpoints are the endpoints of a diameter. Its degree measure is 180. Minor Arc An arc with measure less than 180. Major Arc An arc with measure greater than 180. Angle The union of two rays that have the same endpoint.

10 Sides The rays of the angle. Vertex Where the two rays of an angle meet. Central Angle An angle whose vertex is the center of the circle. Angle Measure Postulate Unique Measure Assumption: Every angle has a unique measure of 0 to 180. Unique Angle Assumption: Given any ray VA and any real number r between 0 and 180, there is a unique angle BVA in each half-plane of VA such that m BVA= r. Zero Angle Assumption: If VA and VB are the same ray, then m AVB= 0. Straight Angle Assumption: If VA and VB are opposite rays, then m AVB=180. Angle Addition Property: If VC is in the interior of m AVC+ m CBV = m AVB. AVB, then Interior of an Angle A nonzero angle separates the plane into two sets of points. If the angle is not straight, the convex set is the interior of the angle. Exterior of an Angle A nonzero angle separates the plane into two sets of points. If the angle is not straight, the nonconvex set is the exterior of the angle. Section 2- Rotations Image The result of applying a transformation to an original figure or preimage.

11 Preimage The original figure in a transformation. Clockwise The direction in which the hands move on a non-digital clock, designated by a negative magnitude. Counterclockwise The direction opposite that which the hands move on a non-digital clock, designated by a positive magnitude. Magnitude In a rotation, the amount that the preimage is turned about the center of rotation, measured in degrees from 180 to 180. Section 3- Adjacent Angles and Vertical Angles Adjacent Angles Two non-straight and non-zero angles with a common side interior to the angle formed by the non-common sides. Angle Bisector A ray is an angle bisector if and only if the ray lies in the interior of the angle and the measure of the two angles created are equal. Angle Measure Postulate Angle Addition Assumption: If angles AVC and CVB are adjacent angles, then m AVC+ m CVB= m AVB. Complementary Two angles the sum of whose measure is 90. Supplementary Two angles the sum of whose measure is 180. Equal Angle Measures If two angles have the same measure, their complements have the same measure. If two angles have the same measure, their supplements have the same measure. Linear Pair Two adjacent angles whose non-common sides are opposite rays. Linear Pair If two angles form a linear pair, then they are supplementary.

12 Vertical Angles Two non-straight and non-zero angles whose sides form two lines. Vertical Angles If two angles are vertical angles, then they have equal measures. Section 4- Algebra Properties Used in Geometry Postulates of Equality Reflexive Property: a = a Symmetric Property: If a =b, then b = a. Transitive Property: If a = b and b = c, then a = c. Postulates of Equality and Operations Addition Property of Equality: If a =b, then a + c = b + c. Multiplication Property of Equality: If a =b, then ac = bc. Postulates of Inequality and Operations Transitive Property: If a b and b c, then a c. Addition Property: If a b, then a + c b + c. Multiplication Property: : If a b and c 0 then ac bc. If a b and c 0, then ac bc. Postulates of Equality and Inequality Equation to Inequality Property: If a and b are positive numbers and a + b = c, then c a and c b.

13 Substitution Property: If a = b, then a may be substituted for b in any expression. Section 5- Justifying Conclusions Proof A sequence of justified conclusions, leading from what is given or known to a final conclusion. Justification A definition, postulate, or theorem which enables a conclusion to be drawn. Section 6- Parallel Lines Transversal A line that intersects two or more lines. Corresponding Angles A pair of angles in similar locations when two lines are intersected by a transversal. Corresponding Angles Postulate Suppose two coplanar lines are cut by a transversal. (1) If two corresponding angles have the same measure, then the lines are parallel. (2) If the lines are parallel, then corresponding angles have the same measure. Slope In the coordinate plane, the change in y-values divided by the corresponding changes in x- y2 y1 values. x x 2 1 Parallel Lines and Slopes Two non-vertical lines are parallel if and only if they have the same slope. Transitivity of Parallelism In a plane, if line l is parallel to line m and line m is parallel to line n, then line l is parallel to line n. Section 7 Size Transformations Concurrent Having a point in common.

14 Point of Concurrency The point at which three or more lines intersect. Transformation A correspondence between two sets of A and B such that: (1) each point in set A corresponds exactly to one point in set B, and (2) each point in set B corresponds to exactly one point in set A. Mapping or Maps Common name for transformations. The transformation maps a preimage onto an image. Size Change When k 0, the transformation under which the image of (x, y) is the point (kx, ky) is the size change (size transformation) of magnitude k and center (0, 0). S k s Parallel Property: Under a size change S k, the line through any two preimage points is parallel to the line through their images. Collinearity Is Preserved: Under S k, the images of collinear points are collinear. Angle Measure Is Preserved: Under S k, an angle and its image have the same measure. Section 8- Perpendicular Lines Perpendicular Two segments, rays, or lines such that the lines containing them form a 90 degree angle. Two Perpendicular If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other. Perpendicular to Parallels In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. Perpendicular Lines and Slopes Two non-vertical lines are perpendicular if and only if the product of their slopes is -1

15 Opposite Reciprocals Two numbers whose product is -1. Section 9- The Perpendicular Bisector Equidistant At the same distance. Bisector of a Segment A line, ray, or segment, which intersects a segment at its midpoint but does not contain the segment. Perpendicular Bisector In a plane, the line containing the midpoint of the segment and perpendicular to the segment. In space, the plane that is perpendicular to the segment and contains the midpoint of the segment. Construction A drawing that is made using only an unmarked straightedge and a compass following certain prescribed rules. Section 1- Reflecting Points Reflecting Line/ Line of Reflection Chapter 4 The line over which a preimage is reflected. Reflection Image For a point P is not on a line m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector of PQ. If P is on m, the reflection image is point P itself. Section 2- Reflecting Figures Reflection Postulate Under a reflection: (a) There is a 1-1 (one-to-one) correspondence between points and their images. (b) Collinearity is preserved. If three points A, B, and C lie on a line, then their images A, B, and C are collinear.

16 (c) Betweenness is preserved. If B is between A and C, then the image B is between the images A and C. (d) Distance is preserved. If A 'B' is the image of AB, A B = AB (e) Angle measure is preserved. If m A' C' E' = m ACE. A' C' E' is the image of ACE, then (f) Orientation is reversed. A polygon and its image, with vertices taken in corresponding order, have opposite orientations. Reflection Image of a Figure The set of all reflection images of the points of the original figure. Figure Reflection If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points. Orientation The order in which the vertices of a polygon are considered, either clockwise or counterclockwise. Section 3 Miniature Golf and Billiards Angle of Incidence When an object hits a surface, the angle formed by the path of the object before contact and the line perpendicular to the surface through the point of contact. Angle of Reflection When an object leaves a hit surface, the angle formed by the path of the object after contact and the line perpendicular to the surface through the point of contact. Section 4- Composing Reflections over Parallel Lines Composite The transformation that maps point P first using the transformation S and a second transformation T, denoted S T, or T(S(P)) Translation/ Slide The composite of two reflections over parallel lines.

17 Direction of a Translation The direction given by the ray from any preimage point to its image point in a translation. Magnitude of a Translation The distance a figure is translated Two-Reflection for Translations If m // l, the translation rm ο rl has magnitude two times the distance between l and m, in the direction from l perpendicular to m. Section 5- Composing Reflections over Intersecting Lines Rotation The composite of two reflections over intersecting lines; the transformation turns the preimage onto the final image about a fixed point (its center) Two-Reflection for Rotations If m intersects l, the rotation rm ο rl has center at the point of intersection of m and l and has magnitude twice the measure of the non-obtuse angle formed by these lines, in the direction from l to m. Section 6- Translations as Vectors Vector A quantity that has both magnitude and direction. Translation Vector A vector that gives the length and direction of a particular translation Initial Point The beginning point of a vector. Terminal Point The endpoint of a vector. Horizontal Component The first component in the ordered pair description of a vector, indicating its magnitude along the x-axis of the coordinate plane. Vertical Component The second component in the ordered pair description of a vector, indicating its magnitude along the y-axis of the coordinate plane.

18 Ordered-Pair Description of a Vector The description of a vector as the ordered pair (a.b) where a is the horizontal component and b is the vertical component. Section 7- Isometries Isometry A transformation that is a reflection or a composite of reflections. Concurrent Two or more lines that have a point in common. Glide Reflection The composite of a reflection and a translation parallel to the reflecting line. Section 8-Transformations and Music None Section 1 When Are Figures Congruent? A-B-C-D Chapter 5 Every isometry preserves angle measure, betweenness, collinearity and distance. Congruent Figures Two figures F and G are congruent figures, written F G, if and only if G is the image of F under an isometry. Congruence Transformation A transformation that is a reflection or composite of reflections. Directly Congruent Figures which are congruent and have the same orientation. Oppositely Congruent Figures which are congruent and have opposite orientation. Equivalence Properties of Congruence Reflexive Property of Congruence: F F

19 Symmetric Property of Congruence: If F G, then G F. Transitive Property of Congruence: If F G, and G H, then F H. Section 2- Corresponding Parts of Congruent Figures Segment Congruence Two segments are congruent if and only if they have the same length. Angle Congruence Two angles are congruent if and only if they have the same measure. Corresponding Parts Angles or sides that are images of each other under a transformation. Corresponding Parts of Congruent Figures (CPCF) If two figures are congruent, then any pair of corresponding parts is congruent. Section 3 One-Step Congruence Proofs- NONE Section 4- Proofs using Transitivity Interior Angle Angles formed by two lines and a transversal whose interiors are partially between the lines. Exterior Angle An angle formed by two lines and a transversal whose interior contains no points between the two lines. Alternate Interior Angles Angles formed by two lines and a transversal whose interiors are partially between the two lines and one different sides of the transversal.

20 Alternate Exterior Angles Nonadjacent angles formed by two lines and a transversal, with interiors inside the two lines and on different sides of the transversal. Same-Side Interior Angles Angles formed by two lines and a transversal, with interior inside the two lines and on the same side of the transversal. Parallel Lines If two lines are cut by a transversal: (1) Alternate interior angles are congruent. (2) Alternate exterior angles are congruent. (3) Same-side interior angles are supplementary. Alternate Interior Angles If two lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. Alternate Exterior Angles If two lines are cut by transversal so that a pair of alternate exterior angles are congruent, then the lines are parallel. Same-Side Interior Angles If two lines are cut by a transversal so that same-side interior angles are supplementary, then the lines are parallel. Section 5- Proofs using Reflections Perpendicular Bisector If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Section 6- Auxiliary Figures and Uniqueness Uniquely Determined When exactly on thing satisfies some given conditions Unique Circle There is exactly one circle (a unique circle) through three given noncollinear points Auxiliary Figure A segment, line, or other figure that is added to a given figure, often to aid in completing proofs.

21 Uniqueness of // s Through a point not on a line, there is exactly one line parallel to the given line. Triangle-Sum The sum of the measures of angles of any triangle is 180. Obtuse Triangle A triangle with an obtuse angle. Right Triangle A triangle with a right angle. Acute Triangle A triangle with an acute angle. Section 7- Sums of Angle Measures in Polygons Quadrilateral- Sum The sum of the measures of the angles of a convex quadrilateral is 360. Polygon-Sum The sum of the measures of the angles of a convex n-gon is (n 2) 180. Exterior Angles of a Polygon An angle is an exterior angle of a polygon if and only if it forms a linear pair with one of the angles of the polygon. Exterior Angle for Triangles In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle. Polygon Exterior Angle The sum of the measures of the exterior angles of a convex n-gon (one per vertex), is 360.

22 Chapter 6 Section 1- Reflection-Symmetric Figures Reflection- Symmetric Figure A figure F for which there is a reflection r m such that r m ( F) = F. Symmetry Line For a figure, a line m such that the figure coincides with its reflection image over m. Flip-Flop (1) If F and G are points and r l ( F) = G, then r l ( G) = F. (2) If F and G are figures and r l ( F) = G, then r l ( G) = F. Segment Symmetry Every segment has exactly two symmetry lines: (1) Its perpendicular bisector, and (2) the line containing the segment. Side-Switching If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle. Angle Symmetry The line containing the bisector of an angle is a symmetry line of the angle. Circle Symmetry A circle is reflection-symmetric to any line through its center. Symmetric Figures If a figure is symmetric, then any pair of corresponding parts under the symmetry are congruent. Section 2- Isosceles Triangles Vertex Angle The angle included by equal sides of an isosceles triangle. Base The side opposite the vertex angle.

23 Base Angles Two angles of an isosceles triangle whose vertices are the endpoints of a base of the triangle. Isosceles Triangle Symmetry The line containing the bisector of the vertex angle of an isosceles triangle is symmetry line for the triangle. Altitude of the Triangle The segment from a vertex perpendicular to the line containing the opposite side. Median of the Triangle A segment from a vertex of the triangle to the midpoint of the opposite side Isosceles Triangle Coincidence In an isosceles triangle, the bisector of the vertex angle, the perpendicular bisector of the base, and the median to the base determine the same line. Isosceles Triangle Base Angles If a triangle has two congruent sides, then the angles opposite them are congruent. Unequal Sides If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the larger side. Converse of the Isosceles Triangle Base Angles If two angles of a triangle are congruent, the sides opposite those angles are congruent. Unequal Angles If two angles of a triangle are not congruent, the side opposite them are not congruent, and the longer side is opposite the larger angle. Section 3 Angles Inscribed in Circles Inscribed Angle An angle in a circle is inscribed if and only if the angle s vertex is on the circle and each of the angle s sides intersects the circle at a point other than the vertex. Intercepted Arc An arc of a circle in the interior of a central angle.

24 Inscribed Angle The measure of an angle inscribed in a circle is half the measure of its intercepted arc. Thales In an inscribed angle intercepts a semicircle, then the angle is a right angle. Section 4- Types of Quadrilaterals Parallelogram A quadrilateral with two pairs of parallel sides. Rhombus A quadrilateral with four sides of equal length. Rectangle A quadrilateral with four right angles. Square A quadrilateral with four sides of equal length and four right angles. Kite A quadrilateral with two distinct pairs of consecutive sides of the same length. Trapezoid A quadrilateral with at least one pair of parallel sides. Bases The parallel sides of a trapezoid Base Angles Two consecutive angles of a trapezoid whose vertices are the endpoints of a base of the trapezoid.

25 Isosceles Trapezoid A trapezoid with a pair of base angles equal in measure. Section 5- Properties of Kites Ends of a Kite The common vertices of the equal sides of the kite. Kite Symmetry The line containing the ends of a kite is a symmetry line for the kite. Symmetry Diagonal The diagonal that connects the ends of the kite. Kite Diagonal The symmetry diagonal of a kite is the perpendicular bisector of the other diagonal and bisects the two angles at the ends of the kite. Rhombus Diagonal Each diagonal of a rhombus is the perpendicular bisector of the other diagonal. Section 6- Properties of Trapezoids Trapezoid Angle In a trapezoid, consecutive angles between a pair of parallel sides are supplementary. Isosceles Trapezoid Symmetry The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and symmetry line for the trapezoid. Corollary An immediate consequence of a theorem. Isosceles Trapezoid In an isosceles trapezoid, the non-base sides are congruent.

26 Rectangle Symmetry The perpendicular bisectors of the sides of the sides of a rectangle are symmetry lines for the rectangle. Section 7- Rotation Symmetry Rotation- Symmetric Figures A figure F for which there is a composite of reflection r ο r such that m l r ( r ( F)) F. m l = Center of Symmetry For a rotation-symmetric figure, the center of a rotation that maps the figure onto itself. N-fold Rotation Symmetry A figure has n-fold rotation symmetry, where n is a positive integer, when a rotation of 360 magnitude maps the figure onto itself, and no larger value of n has this property. n If a figure possesses two lines of symmetry intersecting at a point P, then it is rotationsymmetric with a center of symmetry at P. Section 8- Regular Polygons Regular Polygon A convex polygon whose angles are all congruent and whose sides are all congruent. Equilateral A polygon with all sides of equal length. Equiangular A polygon with all angles of equal measure. Center of a Regular Polygon The point equidistant from all the vertices of the polygon. Center of a Regular Polygon In any regular polygon there is a point (its center), which is equidistant from all of its vertices.

27 Inscribed in a Circle A polygon is inscribed if all of its vertices lie on a circle. Regular Polygon Rotation Symmetry Every regular n-gon possesses n-fold rotation symmetry. Regular Polygon Reflection Symmetry Every regular polygon has reflection symmetry about: (1) each line containing its center and a vertex. (2) each perpendicular bisector of its sides. Section 9- Frieze Patterns Translation Symmetry A figure F has translation symmetry when there is a translation T with nonzero magnitude such that T(F) = F. Frieze Patterns Any pattern in which a single fundamental figure is repeated to form a pattern that has a fixed height and infinite length in opposite directions.

28 Section 1- Drawing Triangles Sufficient Condition Chapter 7 P is a sufficient condition for Q if and only if P implies Q. Included Angle The angle formed by two consecutive sides of a polygon. Section 2- Triangle Congruence SSS Congruence If, in two triangles, three sides of one are congruent to three sides of the other, then the triangles are congruent. Included Angle Of two consecutive sides of a polygon, the angle of the polygon whose vertex is the common point of the sides. SAS Congruence If, in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, then the triangles are congruent. Included Side Of two consecutive angles of a polygon, the side of the polygon which is on both the angles. ASA Congruence If, in two triangles, two angles and the included side of t one are congruent to two angles and the included side of the other, then the two triangles are congruent. AAS Congruence In, in two triangles, two angles and a non-included side of one are congruent respectively to two angles and the corresponding non-included side of the other, then the triangles are congruent. Section 3- Proofs Using Triangle Congruence s

29 Isosceles Triangle Base Angles Converse If two angles of a triangle are congruent, then the sides opposite them are congruent. Section 4- Overlapping Triangles Overlapping Figures Figures which have interior points in common. Section 5- The SSA Condition and HL Congruence SSA Condition Two consecutive sides and the angle NOT included. Hypotenuse The longest side of a right triangle: the side opposite the right angle. Leg Either side of a right triangle that is on the right angle. HL Congruence If, in two right triangles, the hypotenuse and a leg of one are congruent to the hypotenuse and a leg of the other, then the two triangles are congruent. SsA Congruence If two sides and the angle opposite the longer of the two sides in one triangle are congruent, respectively, to two sides and the corresponding angle in another triangle, then the triangles are congruent. Section 6- Tessellations Tessellation A covering of a plane with congruent non-overlapping copies of the same region. Fundamental A region which is used to tessellate a plane.

30 Region Section 7- Properties of Parallelograms Properties of a Parallelogram In any parallelogram: (a) Opposite sides are congruent: (b) opposite angles are congruent: (c) the diagonals intersect at their midpoints. The distance between two given parallel lines is constant. Parallelogram Symmetry Every parallelogram has 2-fold rotation symmetry about the intersection of its diagonals. Section 8- Sufficient Conditions for Parallelograms Sufficient Conditions for a Parallelogram If, in a quadrilateral any of the following apply then the quadrilateral is a parallelogram. (a) One pair of sides is both parallel and congruent (b) Both pairs of opposite sides are congruent (c ) The diagonals bisect each other (d) Both pairs of opposite angles are congruent

31 Section 9- Exterior Angles Exterior Angles An angle is an exterior angle if and only if it forms a linear pair with one of the angles of the polygon. Exterior Angles In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle. Exterior Angle Inequality In a triangle, the measure of an exterior angle is greater than the measure of the interior angle at each of the other two vertices. Unequal Sides If two side of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. Unequal Angles If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

32 Chapter 8 Section 1- Perimeter Formulas Perimeter The length of the boundary of a closed region. Equilateral Polygon Perimeter Formula The perimeter p of an equilateral n-gon with sides of length s is given by the formula p = ns Section 2- Fundamental Properties of Area Square Units A unit used in measuring area. Area Postulate Uniqueness Property: Given a unit region, every polygonal region has a unique area. Rectangle Formula: The area A of a rectangle with dimensions l and w is lw. (A = lw) Congruence Property: Congruent figures have the same area. Additive Property: The area of the union of two non-overlapping regions is the sum of the areas of the regions. Area The number of non-overlapping unit squares or parts of unit squares that can be fit in to a region. Section 3 Areas of Irregular Regions- NONE Section 4- Areas of Triangles Right Triangle Area Formula The area of a right triangle is half the product of the lengths of its legs. 1 A= hb 2

33 Altitude/ Height In a triangle or a trapezoid, the segment from a vertex perpendicular to the line containing the opposite side. Triangle Area Formula The area of a triangle is half the product of a side (the base) and the altitude (height) to that 1 side. A= hb 2 Section 5- Areas of Trapezoids Trapezoid Area Formula The area of a trapezoid equals half the product of its altitude and the sum of the lengths of 1 its bases. A = h( b 1 + b 2 ) 2 Parallelogram Area Formula The area of a parallelogram is the product of one of its bases and the altitude to that bases. A= hb Section 6- The Pythagorean Pythagorean In any right triangle with legs of lengths a and b and hypotenuse of length c, a = b c. Pythagorean Converse If a triangle has sides of lengths a, b, and c, and, triangle. a = b c, then the triangle is a right Section 7- Arc Length and Circumference Circumference The perimeter of a circle, which is the limit of the perimeters of inscribed polygons. Pi C π =, where C is the circumference and d is the diameter of a circle. d

34 Circle Circumference Formula If a circle has circumference C, diameter d, and radius r, then C = πd or C = 2πr. Section 8- The Area of a Circle Sector A region bounded by two radii and an arc of a circle. Circle Area Formula 2 The area A of a circle with radius r is π r.

35 Chapter 9 Section 1- Points, Lines, and Planes in Space Point-Line- Plane Postulate Refer back to Chapter 1 section 7 Flat Plane Assumption: If two points lie in a plane, the line containing them lies in the plane. Unique Plane Assumption: Through three noncollinear points, there is exactly one plane. Intersecting Plane Assumption: If two different planes have a point in common, then their intersection is a line. Section 2- Parallel and Perpendicular Lines and Planes Line-Plane Perpendicularity If a line is perpendicular to two different lines at their point of intersection, then it is perpendicular to the plane that contains those lines. Parallel Planes Planes which have no points in common or are identical. Distance between Parallel Planes The length of a segment perpendicular to the planes with an endpoints in each plane. Section 3- Prisms and Cylinders Surface The boundary of a three dimensional figure. Solid The union of a surface and the region of space enclosed by the surface. Rectangular Solid The union of a box and its interior.

36 Face Any of the polygonal regions that form the surface of the figure. Opposite Faces A pair of faces of a polyhedron whose planes are parallel to each other. Cube A box whose dimensions are all equal. Edges of a box Any side of a polyhedrons faces. Vertices of a Polyherdron Where the edges of a polyhedron meet. Cylindric Solid The set of points between a region (its base) and its translation image in space, including the region and its image. Lateral Surface The surface of the solid excluding its base. Cylindric Surface The set of points between a region (its base) and its translation image in space, including the region and its image. Cylinder The surface of a cylindric solid whose base is a circle. Prism The surface of a cylindric solid whose base is a polygon. Right Cylinder A cylinder/prism formed when the direction of translation of the base is perpendicular to the plane of the base.

37 Oblique A 3-dimensional figure in which the plane of the base is not perpendicular to its axis or to the planes of its lateral surfaces. Lateral Faces Any face other than a base. Regular Prism A right prism whose base is regular polygon. Altitude of a Prism of Cylinder In a prism or cylinder, the distance between the bases. Section 4- Pyramids and Cones Conic Solid The set of points between a given point (its vertex) and all points of a given region (its base), together with the vertex and the base. Pyramid The surface of a conic solid whose base is a polygon. Cone The surface of a conic solid whose base is a circle. Lateral Edge of a Conic Solid Any segment connecting the vertex of the solid to a point on its base. Right Pyramid A pyramid whose base is a regular polygon in which the segment connecting its vertex to the center of its base is perpendicular to the plane of the base. Oblique Pyramid A pyramid whose axis is not perpendicular to its base.

38 Regular Pyramid A right pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. Axis of a Cone The line through the cone s vertex and the center of its base. Right Cone A cone whose axis is perpendicular to the plane of the circular base. Oblique Cone A cone whose axis is not perpendicular to its base. Lateral Surface of a Conic Solid The surface of the solid excluding its base. Lateral Edge of a Conic Solid Any segment connecting the vertex of the solid to a point on its base. Altitude of a Pyramid or Cone In a pyramid or cone, the length of a segment from the vertex perpendicular to the plane of the base. Slant Height of a Regular Pyramid The altitude from the vertex on any one of the lateral faces of the pyramid Slant Height of a Right Cone The length of a lateral edge of the cone.

39 Section 5- Spheres and Sections Sphere The set of points in space at a fixed distance (its Radius) from a point (its center). Great Circle The intersection of a sphere and a plane that contains the center of the sphere. Small Circle The intersection of the sphere and a plane that does not contain the center of the sphere. Hemispheres The half of a sphere on one side of a great circle Plane Section The intersection of a three-dimensional figure with a plane. Section 6- Reflection Symmetry in Space Perpendicular Bisector of a Segment In a plane, the line containing the midpoint of the segment and perpendicular to the segment. In space, the plane that is perpendicular to the segment and contains the midpoint of the segment. Reflection Image of Point A over Plane M If A is not on M, the reflection image is the point B such that M is the perpendicular bisector of AB. If A is on M, the reflection image is point A itself. Congruent Figures A figure F is congruent to a figure G if and only if G is the image of F under a reflection of a composite of reflections. Bilateral Symmetry A space figure has bilateral symmetry if and only if there is a plane over which the reflection image of the figure is the figure itself.

40 Section 7- Viewing Solids and Surfaces Views/ Elevations Planar views of three-dimensional figures given from the top, front, or sides. Section 8- Making Surfaces Polyhedron A three-dimensional surface which is the union of polygonal regions (its faces) and which has no holes. Net A two-dimensional figure that can be folded on its segments or curved on its boundaries into a three-dimensional surface.

41 Chapter 10 Section 1- Surface Areas of Prisms and Cylinders Surface Area The area of the boundary surface of a three-dimensional figure. Right Prism- Cylinder Lateral Area Formula The lateral area, L.A. of a right prism (or right cylinder) is the product of its height h and the perimeter (circumference) p of its base. L. A. = ph Prism-Cylinder Surface Area Formula The surface area, S.A. of any prism or cylinder is the sum of its lateral area L.A. and twice the area B of a base. S. A. = L. A. + 2B Section 2- Surface Areas of Pyramids and Cones Pyramid-Cone Surface Area Formula The surface area, S.A. of any pyramid or cone is the sum of its lateral area L.A. and the area B of its base. S.A. = L.A. + B Regular Pyramid-Right Cone Lateral Area Formula The lateral area, L.A. of a regular pyramid or right cone is half the product of its slant 1 height l and the perimeter (circumference) p of its base. L. A. = lp. 2 Section 3- Fundamental Properties of Volume Volume Postulate Uniqueness Property: Given a unit cube, every polyhedral region has a unique volume.

42 Box Volume: The volume V of a box with dimensions l, w, and h is found by the formula V = lwh. Congruence Property: Congruent figures have the same volume. Additive Property: The volume of the union of two non-overlapping solids is the sum of the volume of the solids. Cube Volume Formula The volume V of a cube with edge s is 3 s. 3 V = s Section 4 Multiplication, Area, and Volume- NONE Section 5- Volumes of Prisms and Cylinders Volume Postulate Cavalieri s Principle: Let I and II be two solids included between parallel planes. If every plane P parallel to the given planes intersects I and II in sections with the same area, then Volume(I) = Volume (II) Prism-Cylinder Volume Formula The volume V of any prism of cylinder is the product of its height h and the area B of its base. V = Bh Section 6 Organizing Formulas-NONE Section 7- Volumes of Pyramids and Cones Pyramid-Cone Volume Formula The volume V of any pyramid of cone equals 3 1 the product of its height h and its base area B. 1 V = Bh 3

43 Section 8- The Volume of a Sphere Sphere Volume Formula The volume V of any sphere is 4 V = π r π times the cube of its radius r. 3 Section 9- The Surface Area of a Sphere Sphere Surface Area Formula The total surface area S.A. of a sphere with radius r is 2 S.A. = 4π r 2 4π r.

44 Chapter 11 Section 1- The Logic of Making Conclusions Law of Detachment From a true conditional p => q and a statement of given information p, you may conclude q. Law of Transitivity If p => q and q => r are true, then p => r is true. Section 2- Negations Negation A statement (called not-p) that is true whenever statement p is false and is false whenever statement p is true. Contrapositive A conditional resulting from negating and switching the antecedent and consequent of the original conditional. Law of the Contrapositive A conditional (p => q) and its contrapositive (not-p => not-q) are either both true of both false. Section 3- Ruling Out Possibilities Law of Ruling Out Possibilities When statement p of statement q is true, and q is not true, then p is true. Section 4- Indirect Proofs Direct Reasoning Reasoning (proofs) using the Law of Detachment and /or the Law of Trasitivity. Indirect Reasoning Reasoning (proofs) using the Law of the Contrapositive, the Law of Ruling Out Possibilities, or the Law of Indirect Reasoning. Contradictory Two statements that cannot both be true at the same time.

45 Law of Indirect Reasoning If valid reasoning from a statement p leads to a false conclusion, then p is false. Section 5- Proofs with Coordinates Convenient Location A location in which its key points are described with the fewest possible number of variables. Section 6- The Distance Formula Distance Formula The distance d between two points ( x 1, y1) and ( x 2, y2 ) in the coordinate plane is given by the formula d = ( x y x1 ) + ( y2 1) Section 7- Equations for Circles Equation for a Circle The circle with center (h,k) and radius r is the set of points (x,y) satisfying ( x h) + ( y k) = r. Section 8- Means and Midpoints Mean The sum of a set of numbers divided by the number of numbers in the set. Number Line Midpoint Formula On a number line, the coordinate of the midpoint of the segment with endpoints a and b is a+ b. 2 Coordinate Plane Midpoint Formula In the coordinate plane, the midpoint of the segment with endpoints ( x 1, y1) and ( x 2, y2 ) is x 1+ x2 y1+ y2 (, ). 2 2

46 Midpoint Connector The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.