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Transcript
Geometry Vocabulary Word Definition Examples Acute angle An angle that measures between 0 and 90 degrees 0° < x < 90° Acute triangle All angles in the triangle are acute A 40 - 60 - 80 triangle is acute Adjacent angles Two coplanar angles with a common side, a common vertex, and no common interior points Adjacent arcs Two arcs in the same circle that have exactly one point in common. Two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a Alternate description of the location of the angles); alternate interior angles interior angles are congruent if the lines intersected by the transversal are parallel. Angle The shape formed by two rays (called sides of the angle) with the same endpoint (called the vertex of the angle). In geometry an angle can be defined by the vertex or by the rays and vertex. Angle bisector A ray that divides an angle into two congruent (equal) angles Exactly half the circle is called a semicircle. Less than half is a minor arc and more than half is a major arc. Arc Part of a circle Arc length The length in linear measure of an arc of a circle; the product of the ratio of the arc measure and 360º and the circumference of the circle Angle of Depression The angle between the An angle from the horizontal down to viewer and something below a line of sight them Angle of Elevation An angle from the horizontal up to a line of sight Bisector (of a line segment) A line, segment, ray, or plane that intersects a segment at its midpoint The angle between the ground and something above the ground Bisector (of an A line or ray that divides the angle into two congruent angles. angle) Central angle An angle whose vertex is the center of a circle Centroid The point in a triangle where the medians of the triangle intersect. Chord A line segment whose endpoints lie on the circle. If a chord goes through the center of a circle, it is a diameter. Circle The set of all points in a plane that are a constant distance of r from a given point called the center. If the center is at (h,k) and the radius is r, the equation of the circle is (x-h)2 + (y-k)2 = r2 Circle Graph A graph that represents the frequency of data as slices of a circle. The size of each slice represents the frequency, usually recorded as a percent. Circumcenter The point in a triangle where the perpendicular bisectors of the sides of the triangle intersect. Collinear Points that lie on the same line Compass A tool used to draw circles and parts of circles Complementary Two positive angles whose measures angles add to 90 degrees (x-2)2 + (y-3)2 = 52 is a circle with center at (2,3) and a radius of 5 Points that all line up A + B = 90° Concave polygon When a diagonal contains points outside the polygon. Conditional statement A statement that can be written in the form “If p, then q.” p is the hypothesis and q is the conclusion. Symbolically, if p, then q can be written as p→q. Congruent Objects that are the same size and the same shape. The symbol for congruent is an equal sign with a squiggle over it. Conjecture The conclusion reached in a math statement based on reasoning Construction Using only a compass and a straight edge to draw a geometric figure. Contrapositive In the contrapositive of a conditional statement, the hypothesis and conclusion are both reversed and negated. “If p, then q.” becomes “If not q, then not p.”. The contrapositve has the same truth value as the original statement. Converse In the converse of a conditional statement, the hypothesis and conclusion are reversed. “If p, then q.” becomes “If q, then p.”. Convex polygon No diagonal contains points outside the polygon Coplanar Points and lines that are in the same plane Corresponding angles (parallel lines) Two angles that lie on the same side of a transversal, in corresponding positions with respect to the two lines that the transversal intersects. Corresponding angles (in polygons) Angles that are in the same position in different plane figures Corresponding parts Matching sides and angles in a polygon. Corollary A statement that follows directly from a theorem. Cosine The ratio of the length of the side side adjacent adjacent to an angle and the divided by hypotenuse in a right triangle hypotenuse Deductive reasoning A method of reasoning logically from given facts, rules, definitions, or properties to a conclusion. Diameter In a circle, any segment that contains the center of the circle and has both endpoints on the circle. Dilation A transformation in which a figure is made larger or smaller, but not moved. Distance Formula Using the Pythagorean Theorem to find the distance between two points on a coordinate grid) Equiangular polygon A polygon where all the angles are congruent Equiangular triangle A triangle where all the angles are All angles equal congruent Equilateral polygon A polygon where all the sides are congruent All sides equal Equilateral triangle A triangle where all the sides are congruent All sides equal Euclidean geometry A geometric system based on the five postulates of Euclid 1. A straight line can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All angles equal 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate (the parallel postulate) Euler Line In geometry, the centroid, circumcenter and orthocenter of any triangle always lie on a straight line, called the Euler line. Exterior angle In a polygon, the angle formed by a side and an extension of a side. Foundation drawing Drawing that shows the base of an object (its foundation) and the height of each part. Geometric Mean The geometric mean of two numbers a and b is the square root of a times b Geometry The study of lines, shapes, measurements, angles, and figures in space based on defined properties. Plane geometry studies shapes on flat surfaces, like your paper or the chalkboard. Spherical geometry studies shapes on a sphere, like a ball or the earth. Hypotenuse In a right triangle, the side opposite the right angle - the longest side in a right triangle Hypothesis In a conditional statement (if _ then_), the statement that immediately follows the if. If and only if (conditional statements) In an equivalence statement, the words if and only if may be represented by the short symbol iff. Then the definition of an equivalence statement is written as follows: p iff q = (p implies q) and (q implies p). Here the first implication means that when p is true, q must be true, and we cannot have p true and q false. The second implication means that when q is true, p must be true, and we cannot have q true and p false. Therefore, in an equivalence statement the only The assumption you start with possibilities are: (1) p and q both true, and (2) p and q both false. Then p and q are said to be equivalent. Image The position of a figure after a transformation. Incenter The point in a triangle where the angle bisectors of the triangle intersect. For the pattern 2, 4, 6, 8 you can induce that since they are all even numbers, the next number must be 10 Inductive Reasoning A method used to reach conclusions based on an observed pattern. Intersect Lines intersect when they cross. The point where they cross is called the intersection. Isometric drawing The drawing of a three-dimensional object that shows the corners. Isosceles trapezoid A trapezoid whose nonparallel sides are congruent. The top and bottom are parallel, the sides are the same size, and the diagonals are equal. Isosceles triangle A triangle where at least two sides are congruent Kite A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent Law of Detachment Suppose that the statements p and p implies q are both true. Then we may write: p and (p implies q). By the definition of the implication, this is: p and (notp or q). By the distributive law, we now have: (p and not-p) or (p and q). By the law of contradiction p and not-p is false, so p and q must be true. This shows we may conclude that q is true, because we have already supposed that p is Two or more sides equal true. We may summarize this result as follows: From p and (p implies q) we conclude q. Law of Syllogism Suppose the statements p implies q and q implies r are both true. Then we may write: (p implies q) and (q implies r). The first implication means that when p is true, q must also be true and we cannot have p true and q false. The second implication means that when q is true r must also be true, and we cannot have q true and r false. These results show that when p is true r must also be true, and we cannot have p true and r false. In other words: p implies r. We may summarize this result as follows: From (p implies q) and (q implies r) we conclude (p implies r). Legs of a right triangle The sides that form the right angle in a right triangle Line A series of points that extends forever in two directions. A line is uniquely defined if you know two points on the line. Line segment Part of a line consisting of two endpoints and all the points between them. Linear pair two adjacent angles whose non-common sides are opposite rays; the sum of the measures of the angles in a linear pair is 180 º <1 and <2 comprise a linear pair. Median geometry In a triangle, a line drawn from the midpoint of a side to the vertex opposite. Every triangle has three medians. Midpoint The point that divides a line into two equal pieces. For the line with end points at A(x1,y1) and B(x2,y2), the midpoint is located at: x1 + x2 y1 + y2 , 2 2 The midpoint of a line with ends at (2,5) and (4,7) is at (3,6) Negation The opposite of a statement: If a statement is represented by p - Today is Monday p, then the negation is not p. not p - Today is not Monday. A polygon with 5 sides is a pentagon A polygon with 42 sides is a 42-gon N-gon A polygon with n sides. Oblique triangle Any triangle that is not a right triangle Obtuse angle An angle that measures between 90 and 180 degrees Obtuse triangle A triangle with one obtuse angle Opposite rays Two collinear rays that share the same endpoint and together form a straight line Orthocenter The point in a triangle where the altitudes of the triangle intersect. Orthographic drawing A drawing that represents a three dimensional object by showing separate drawings for the front, top and right side views Parallel lines Lines that are always the same distance apart, they never cross. Parallelogram A quadrilateral where the Rectangles, squares, and diamonds are opposite parallelograms sides are parallel Pascal's Triangle A pattern for finding the coefficients of the terms 90° < x < 180° One angle bigger than 90° Lines that y = 3x + 1 and y = 3x - 5 are have the parallel because they both same have slope = 3 slope 1 1 1 1 2 1 1 3 3 1 in a binomial expansion. It is also used in probability. Perpendicular bisector A line, segment, or ray that is perpendicular to the original segment at its midpoint Perpendicular lines Lines that cross at right angles. The product of the slopes is -1. Perspective drawing Drawing that makes a two-dimensional image look like a three-dimensional object Pi Π π Most frequently used Greek letter, it represents the ratio of diameter to circumference in math problems about circles. It also represents an angle measure in radians equal to 180° Plane A flat surface that extends forever in all directions. Planes have no thickness. Usually drawn as a shaded parallelogram. Platonic solids The five regular polyhedra: tetrahedron, hexahedron, octahedron, dodecahedron, or icosahedron Point A location in space that has no size. Any point can be uniquely identified by a set of coordinates (x, y). Polygon A closed plane figure with at least three sides Postulate An accepted statement "Two points define exactly one of fact that can be straight line" is a postulate y = 2x + 1 and y = -½x - 5 are perpendicular because 2 • (-½) = -1 A triangle is a 3-sided polygon used to prove theorems Pre-Image The position of a figure before a transformation Proof A valid argument in which all of the premises are true Protractor A device used to measure the size of an angle in degrees. Pythagorean theorem In a right triangle, the sum of the squares of the lengths of the two sides (legs a and b) is equal to the square of the length of the hypotenuse (c). a2 + b2 = c2 Quadrilateral A geometric figure with four sides and four angles A rectangle is quadrilateral Radius In a circle, any segment that has one endpoint at the center of the circle and the other endpoint on the circle Ray The part of a line consisting of one endpoint and all the points extending forever in one direction Rectangle A parallelogram with four right angles Regular polygon A polygon that is both equilateral and equiangular. For each exterior angle of a triangle, the two non- Remote interior angles adjacent angles Rhombus A parallelogram with A diamond is a rhombus four congruent sides Right Angle An angle that measures 90° Right Triangle A triangle that has one 90° angle. The other two angles will add up to 90° Same-side interior angles Two angles that lie on the same side of a transversal and between the lines cut by the transversal, in corresponding positions with respect to the two lines that the transversal intersects. x = 90° Scalar A quantity that represents a magnitude. It can be represented by a real number. Scale Scale is the ratio of any length in a scale drawing to the corresponding actual length; the lengths may be in different units Scalene triangle A triangle with no sides congruent Secant A line, ray, or segment that intersects a circle at two points (i.e. that contains a chord). A secant to a sphere is a line, ray, or segment that intersects a sphere at two points. Sector of a circle The portion of the interior of a circle intercepted by a central angle. Segment Part of a line consisting of two endpoints and all the points between them Semicircle Exactly half a circle Similar Figures Figures that have the same shape, but not the same size. Same angles, proportional sides Similar Triangles Triangles that have the same angles, but not necessarily the same size. Same shape Similarity ratio The ratio of the lengths of corresponding sides in a similar polygon. Sine The ratio of the length of the side opposite an angle and the hypotenuse in a right triangle Skew lines Two lines that are not in the same plane. They do not intersect and they are not parallel. Space The set of all points. Square A parallelogram with four All sides and angles congruent sides and four right equal angles Straight angle An angle that measures 180°, a straight line. All sides are a different length Like a slice of a round pie side opposite divided by hypotenuse x = 180° Straightedge A ruler that doesn't have any markings on it used to draw lines where the exact length doesn't matter. Supplementary angles Two positive angles whose measures add to 180 degrees. When adjacent (sharing a side) supplementary angles make a straight line. Tangent (to a circle) A line that intersects a circle in exactly one point. The ratio of the length of the Tangent side opposite an angle and the (trigonometric function) side adjacent to the angle in a right triangle. A + B = 180° side opposite divided by side adjacent Theorem A conjecture that is proven. In geometry, properties, postulates and theorems are used to prove other theorems Transversal A line that intersects two or more other lines in the same plane at different points. Trapezoid A quadrilateral with exactly one pair of parallel sides. Triangle A polygon with three sides and three angles Undefined terms Words, usually easily understood, that are not formally explained or defined. Vertex The point on a triangle where the sides come together. The vertices of triangles are usually labeled with capital letters. Two sides are parallel, other sides are not. In geometry, point, line, and plane are undefined terms <> Vertical angles Two angles whose sides are opposite rays Volume The measure of the amount of space enclosed by a threedimensional figure. The volume of a cube is side X side X side