* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Math Extra Credit
Noether's theorem wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Multilateration wikipedia , lookup
Perspective (graphical) wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Perceived visual angle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Chapter 1: Section 1.8 Definitions: 1. Betweenness: A number is between two others if it’s greater than one of them and less than the other. Postulates: 1. Distance Postulate: There are 3 parts to the distance postulate a. Uniqueness Property: On a line there is a unique distance between two points. b. Distance Formula: If two points on a line have coordinates x and y. c. Additive Property: If B is on line segment AC ( -- ), then AB + BC = AC Chapter 2: Section 2.7 Definitions: 1. The Triangle Inequality: Any side of a triangle is shorter than the other two sides combined. a<b+c b<a+c c<a+b Chapter 3: Section 3.1 Definitions: 1. Angle: The union of two rays that have the same endpoint. 2. Sides: The sides of an angle are the two rays that form it. 3. Vertex: the vertex of the angle is the common endpoint of the two rays. 4. Interior: The convex set is inside of the angle. 5. Exterior: The convex set is outside of the angle. 6. Measure: Indicates the amount of openness of the interior of the angle. 7. Degree: A unit of measure used for the measure of an angle, arc, or rotation 8. Bisector: A ray in the interior of an angle that divides angle into two angles of equal measures. Postulates: 1. Unique Measure Assumption: Every angle has a unique measure from 0 degrees to 180 degrees. 2. 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 line VA (<-->) such that m<BVA = r 3. Zero Angle Assumption:If ray VA (→) and ray VB (→) are the same ray then m<AVB = 0 4. Straight Angle Assumption: If ray VA (→) and ray VB (→) are opposite rays, then m<AVB = 180 5. Angle Addition Property: If ray VC (→) (except for point V) is in the interior of <AVB, then m<AVC + m<CVB = m<AVB. Section 3.2 Definitions: 1. Central angle of a circle: An angle whose vertex is the center of the circle. 2. Minor arc: The points of circle O that are on or in the interior of angle AOB 3. Major arc: The points of circle O that are on or in the exterior of angle AOB. 4. Semicircle: An arc of a circle whose endpoints are the endpoints of a diameter of the circle. 5. Degree measure of a minor arc or semicircle: The measure of the central angle AOB 6. Degree measure of a major arc (ACB): 360 - (measure of minor arc AB) 7. Concentric: Two or more circles that lie in the same plane and have the same center. 8. Images: The result of applying a transformation to an original figure or preimage. 9. Preimages: The original figure in a transformation. 10. Clockwise direction: The direction in which the hands move on the nondigital clock, designated by negative magnitude. 11. Counterclockwise direction: The direction opposite that which the antecedent is true but the consequent is false. An example which shows conjecture to be false. 12. Magnitude: In a rotation, the amount that the preimage is turned about the center of rotation, measured in degrees from -180 degrees (clockwise) to 180 degrees (counterclockwise), positive/negative measure angle POP, where P is the image of P under the rotation and O is its center Section 3.3 Definitions: 1. If m is the measure of an angle, then the angle is: a. b. c. d. e. zero if and only if m = o acute if and only if 0 < m < 90 right if and only if m = 90 obtuse if and only if 90 < m < 180 straight if and only if m = 180 2. Complementary Angles: When two measures add up to 90° 3. Supplementary Angles: When two measures add up to 180° 4. Adjacent Angles: Two non straight and nonzero angles with a common side interior to the angle formed by the noncommon sides. 5. Linear Pair: Two adjacent angles if and only if their noncommon sides are opposite rays. 6. Vertical Angles: Two non-straight angles if and only if the union of their side is two lines. Theorem: 1. Linear Pair Theorem: If two angles for a linear pair, then they are supplementary. 2. Vertical Angles Theorem: If two angles are vertical angles, then the have equal measures. Postulates: 1. Angle Addition Postulate: <ABD (Angle ABD) + <DBC (Angle DBC) = <ABC (Angle ABC) Section 3.4 Postulates: 1. Postulates of Equality: a. Reflexive Property: a = a b. Symmetric Property: If a = b; b = a c. Transitive Property of Equality: If a = b and b = c, then a = c 2. Postulates of Equality and Operations: a. Addition Property: You can add/subtract a number from both sides of an equation. b. Multiplication Property: You can multiply/divide an equation by the same number on both sides (except 0). c. Substitution Property: If a = b, you can replace a with b anywhere. 3. Postulates of Inequality and Operations: a. Transitive Property: If a < b and b < c, then a < c. b. Addition Property: If a < b, then a + c < b + c. c. Multiplication Property: If a < b and c > 0, then ac < bc. If a < b and c < 0 , then ac > bc. 4. Postulates of Equality and Inequality: a. Equation of Inequality Property: If a and b are positive numbers and a + b = c, then c > a and c > b. b. Substitution Property: If a = b, then a may be substituted for b in any expression. Section 3.5 Definition: 1. Proof argument: Contains a conclusion and a justification. 2. Justification: a reason why a statement is true. ● There are 3 types of justifications: postulates/properties, definitions, and theorems. Section 3.6: Definitions: 1. Transversal: A line that intersects 2 others. 2. Corresponding angles: A pair of angles in similar locations when two lines are intersected by a transversal. 3. Slope: In the coordinate plane, the change in y-value divided y the corresponding changes in x-values. Postulates 1. Corresponding Angle Postulate: Two (coplanar) lines are parallel if and only if corresponding angles created by a transversal are equal. Theorem: 1. Parallel Lines and Slopes: Two nonvertical lines are parallel if and only if they have the same slope. 2. Transactivity of Parallelism: In a plane if l // m and m // n, the l // n. Section 3.7 Definitions: 1. Perpendicular: If 2 lines are perpendicular, they form a 90° angle. 2. Parallel: 2 lines are parallel if they never intersect Theorem 1. Two perpendiculars: If 2 coplanar line l and m are each perpendicular to the same line, then they are parallel to each other. 2. Perpendicular to Parallels: In a plane, if a line is perpendicular to one of 2 parallel lines, then it’s also perpendicular to the other. 3. Perpendicular Lines and Slopes: 2 nonvertical lines are perpendicular if and only if the product of their slopes is -1. 4. Parallel Lines and Slopes: The slope of parallel lines are equal. Section 3.8 Definitions: 1. Bisector of a segment: Has 2 equal pieces even though it does not create a 90° angle. 2. perpendicular bisector: Splits a segment into two segments and creates a 90° angle. 3. Construction: A drawing which is made using only an unmarked straightedge and a copass following certain prescribed rules. 4. Unmarking straightedge: An instrument with a straight edge (straight line such as a ruler etc.) 5. compass: An instrument for drawing circles. 6. Algorithm: A finite sequence of steps leading to a desired end. Chapter 4: Section 4.1 Definitions: 1. preimage: preimage of an object is the object we are reflecting. 2. reflecting line: The line over which a preimage is reflected. Also called line of reflection or mirror. 3. reflection image of P overline m: If p is not on m, the reflection image of p is the point Q such that m is the perpendicular bisector of line segment PQ. If P is on m, the reflection image is point P itself. 4. transformation: a correspondence between 2 sets of points such that: a. each point in the preimage set has a unique image. b. each point in the image has exactly one preimage. 5. mapping: a transformation. Section 4.2 Definition: 1. Reflection image of a figure: The set of all reflection images of the points of the original figure. Postulates: 1. Reflection: Reflections preserve: a. angle measure b. betweenness c. collinearity d. distance e. orientation Theorem: 1. 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. Section 4.4 Definition: 1. Composite: The composite of a first transformation S and a second transformation T; denoted T°S is the transformation that maps each point P onto T (S(P)) 2. translation or slide: reflecting over 2 parallel lines 3. direction: the direction of a translation is which way it travels a. Perpendicular to the lines and towards the second line. 4. Magnitude: the magnitude of a translation is the distance it travels. a. TWICE the distance between parallel lines Theorem: 1. Two-Reflection Theorem for Translation: If m // l, the translator r (lowercase m) composite (°) r (lowercase l) has magnitude two times the distance between l and m, in the direction from l perpendicular to m Section 4.5 Definition: 1. Rotation: The composite of two reflections over intersecting lines; the transformation “turns” the preimage onto the final image about a fixed point (its center). Also called turn. Theorem: 1. Two-Reflection Theorem for Rotations: If m intersects l, the rotation r (lowercase m) composite ( ° ) r (lowercase l) has center at the point of intersection of m and l and has magnitude twice the measure of non-obtuse angle formed by these lines, in the direction from l to m. Section 4.7 Definition: 1. isometry: A transformation that is a reflection or a composite of reflections. Also called congruence transformation or distance-preserving transformation. a. 3 types of isometries: 1 Reflection = Reflection; 2 Reflections = Translation & Rotation; Reflection + Translation = glide reflection. 2. concurrent: Two or more lines that have a point in common Chapter 5: Section 5.1 Definition: 1. corresponding parts: Angles or sides that are images of each other under a transformation. Theorem: 1. Corresponding parts of congruent figures: If two figures are congruent, then any pair of corresponding parts is congruent. 2. A-B-C-D: Every isometry preserves Angle measure, Betweenness, Collinearity (lines), and Distance (length of segments). Section 5.2 Theorem: 1. Equivalence Properties of congruent: For any figures F, G, H: a. Reflexive: F congruent to F b. Symmetric: If F congruent G, then G congruent F c. Transitive: If F congruent G and G congruent H, the F congruent H 2. Segment Congruence: Two segments are congruent if and only if they have the same length. 3. Angle Congruence: Two angles are congruent if and only if they have the same measure. Section 5.4 Definition: 1. two-column form: A form of written proof in which the conclusion are written in one column, the justifications beside them in a second column. 2. interior angles: Angles formed by two lines and a transversal whose interiors are partially between the lines; the angles of a polygon. 3. exterior angles: An angle formed by two lines and a transversal whose interior contains no points between the two lines. An angle which forms a linear pair with an angle of a given polygon. 4. alternate interior angles: Angles formed by two lines and a transversal whose interiors are partially between the two lines an on different sides of transversal. 5. alternate exterior angles: Angles formed by two lines and a transversal whose interior are not between the two lines and are on different sides of the transversal. Theorem: 1. Lines ⇒AIA congruent (Alternate Interior Angles Theorem): If two parallel lines are cut by a transversal , then alternate interior angle are congruent. 2. AIA congruent ⇒// lines: If two lines are cut by a transversal and form congruent alternate interior angles, then the lines are parallel. Section 5.5 Definition: 1. equidistant: At the same distance Theorem: 1. Perpendicular bisector: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Section 5.6 Definition: 1. uniquely determined: One and exactly one. Example: The line between 2 points, the perpendicular bisector of a segment, the measure of an angle, and the midpoint of a segment. 2. auxiliary figure: gives help or assisting. Auxiliary figures are objects we add to a diagram to help us prove something. Theorem: 1. Uniqueness of Parallels: Through a point not on a line, there is exactly one line parallel to the given line. Section 5,7 Theorem: 1. Triangle-sum: The sum of the measures of tje amg;es pf a triangle is 180 degree 2. quadrilateral-sum: The sum of the measures of the angles of a convex quadrilateral is 360 degrees 3. polygon-sum: The sum of the measures of the angles of a convex n-gon is (n - 2) x 180 degrees.