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Geometry Vocabulary Chapter 1 Undefined Terms: words like point, line, and plane that are words that have no formal definitions, but there is agreement about what they mean. Point: has no dimension, represented by a dot. Line: has one dimension. It is represented by a line with two arrowheads, but extends without end. Through any two points there is exactly one line. Plane: has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end. Through any three points not on the same line, there is exactly one plane. Collinear Points: points that lie on the same line. Coplanar Points: points that lie on the same plane. Defined Terms: terms that can be described using known words such as point and line. Segment: the line segment AB, or segment AB, (written as π΄π΅) consists of the endpoints A and B and all points on β‘π΄π΅ (line β‘π΄π΅ ) that are between A and B. Ray: the ray AB (written as π΄π΅ ) consists of the endpoint A and all points on β‘π΄π΅ that lie on the same side of A as B. Opposite Rays: If point C lies on β‘π΄π΅ between A and B, then πΆπ΄ and πΆπ΅ are opposite rays. Intersection: between two figures, is the set of points the figures have in common. Postulate: a rule that is accepted without proof, also called an axiom. Coordinate: the real number the corresponds to a point on a line. Distance: between points A and B is the absolute value of the difference of the coordinates A and B Between: when three points are collinear, you can say that one point is between the other two. Congruent Segments: line segments that have the same length. Midpoint: is the point that divides the segment into two congruent segments. Segment Bisector: a point, ray, line segment, or plane that intersects the segment at its midpoint. Angle: consists of two different rays with the same endpoint. Sides: the rays of the angle Vertex: the endpoint of the angle Measure of an Angle: the number of degrees the angle opens to. Acute Angle: classification of an angle when the measure of the angle is between 0° and 90°. Right Angle: classification of an angle when the measure of the angle is exactly 90°. Obtuse Angle: classification of an angle when the measure of the angle is between 90° and 180°. Straight Angle: classification of an angle when the measure of the angle is exactly 180°. Congruent Angles: when angles have the same measure. Angle Bisector: a ray that divides an angle into two angles that are congruent. Complementary Angles: when the sum of two angle measures is equal to 90°. Supplementary Angles: when the sum of two angle measures is equal to 180°. Adjacent Angles: two angles that share a common vertex and side, but have no common interior points. Linear Pair: two adjacent angles are this is their non-common sides are opposite rays. The angles in a linear pair are supplementary angles. Vertical Angles: two angles are this if their sides form two pairs of opposite rays Chapter 2 Conjecture: an unproven statement that is based on observations. Inductive Reasoning: when you find a pattern in specific cases then write a conjecture for the general case. Counterexample: a specific case for which a conjecture is false. Conditional Statement: a logical statement that has two parts, a hypothesis and a conclusion. If-Then Form: when a conditional statement is written with the if part having the hypothesis and the then part having the conclusion. Negation: the opposite of the original statement. Converse Statement: the conditional statement with the hypothesis and the conclusion switched. Inverse Statement: negation of both the hypothesis and the conclusion of the conditional statement. Contrapositive Statement: write the converse statement then negate both the hypothesis and the conclusion. Perpendicular Lines: when two lines intersect to form a right angle. Biconditional Statement: a statement that contains the phrase βIf and only ifβ. Any definition can be written as a biconditional statement Deductive Reasoning: uses facts, definitions, accepted properties, and the laws of logic to form a logical argument. Line Perpendicular to a Plane: a ling is this if and only if the line intersects the plane in a point and is perpendicular to every line in the plan that intersects it at that point. Proof: a logical argument that shows a statement is true. Two-Column Proof: has numbered statements and corresponding reasons that show an argument in a logical order. Theorem: is a statement that can be proven. Once a theorem is proven, it can be used in other proofs. Chapter 3 Parallel Lines: lines that do not intersect and are coplanar. Skew Lines: lines that do not intersect and are not coplanar. Parallel Planes: two planes that do not intersect. Transversal: is a line that intersects two or more coplanar lines at different points Corresponding Angles: two angles are this is they have corresponding positions. Alternate Interior Angles: two angles are this if they lie between the two lines and on opposite sides of the transversal. Alternate Exterior Angles: Two angles are this if they lie outside the two lines and on opposite sides of the transversal. Consecutive Interior Angles: two angles are this if they lie between the two lines and on the same side of the transversal. Paragraph Proof: a proof written in paragraph form using sentences with words to explain the logical flow of the argument. Slope: the ratio of the vertical change to the horizontal change between any two points on the line. Slope-Intercept Form: the general linear equation, π¦ = ππ₯ + π, where m is the slope and b is the y-intercept. Standard Form: another form of a linear equation, written as π΄π₯ + π΅π¦ = πΆ, where A and B are not both zero. Distance from a Point to a Line: the length of the perpendicular segment from the point to the line. Chapter 4 Triangle: a polygon with three sides. A triangle with vertices A, B, and C is called βtriangle ABCβ or ββπ΄π΅πΆβ. Interior Angles: these are the original angles when the sides of a polygon are extended. Exterior Angles: these angles form linear pairs with the interior angles when the sides of a polygon are extended. Corollary to a Theorem: a statement that can be proved easily using the theorem. Congruent Figures: all the parts of one figure are congruent to the corresponding parts of the other figure. Corresponding Parts: a pair of sides or angles that have the same relative position in two congruent or similar figures. Legs of a Right Triangle: the sides adjacent to the right angle. Hypotenuse: the side opposite the right angle. Flow Proof: uses arrows to show the flow of a logical argument Legs of an Isosceles Triangle: exactly two congruent sides. Vertex Angle: the angle formed by the legs. Base: the third side of the isosceles triangle. Base Angles: the angles adjacent to the base. Transformation: is an operation that moves or changes a geometric figure in some way to produce a new figure Image: the new figure produced by the operation Translation: moves every point of the figure the same distance in the same direction. Reflection: uses a line of reflection to create a mirror image of the original figure. Rotation: turns a figure about a fixed point, the center of rotation. Congruence Transformation: changes the position of the figure without changing its size or shape. Chapter 5 Midsegment of a Triangle: a segment that connects the midpoints of two sides of the triangle. Coordinate Proof: involves placing geometric figures in a coordinate plane. Perpendicular Bisector: a segment, ray, line, or plane that is perpendicular to a segment at its midpoint. Equidistant: when a point is the same distance from each figure. Concurrent: when three or more lines, rays, or segments intersect in the same point. Point of Concurrency: the point of intersection of the lines, rays, or segments. Circumcenter: the point of concurrency of the three perpendicular bisectors of a triangle. Incenter: the point of concurrency of the three angle bisectors of a triangle. Median of a Triangle: a segment from a vertex to the midpoint of the opposite side. Centroid: the point of concurrency inside the triangle. Altitude of a Triangle: the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. Orthocenter: the point at which the lines containing the three altitudes of a triangle intersect. Indirect Proof: you start by making the temporary assumption that the desired conclusion is false. Then show that this assumption leads to a logical impossibility, this proves the original statement true by contradiction. Chapter 6 π Ratio of a to b: if a and b are two numbers or quantities and b β 0, the ratio is π. Proportion: an equation that states two ratios are equal. π π Means: the numbers b and c of the proportion: π = π π π Extremes: the numbers a and d of the proportion: π = π . π π Geometric Mean: of two positive numbers a and b is the positive number x that satisfies π = π. So, π2 = ab and x = βππ. Scale Drawing: is a drawing that is the same shape as the object it represents. Scale: a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object. Similar Polygons: corresponding angles are congruent and corresponding side lengths are proportional. Scale Factor: when two polygons are similar, this is the ratio of the lengths of two corresponding sides. Dilation: a transformation that stretches or shrinks a figure to create a similar figure. A type of similarity transformation. Center of Dilation: in a dilation, a figure is enlarged or reduced with respect to a fixed point. Scale Factor of a Dilation: the ratio of a side length of the image to the corresponding side length of the original figure. Reduction: if the dilation is 0 < π < 1, where k is the scale factor. Enlargement: if the dilation is π > 1, where k is the scale factor. Chapter 7 Pythagorean Triple: a set of three positive integers a, b, and c that satisfy the equation π 2 = π2 + π 2 . Trigonometric Ratio: a ratio of the lengths of two sides in a right triangle. Tangent: the ratio of the lengths of the legs in a right triangle is constant for a given angle measure. Sine: trigonometric ratios of the length of leg opposite of length of hypotenuse. Cosine: trigonometric ratio of the length of leg adjacent to length of hypotenuse. Angle of Elevation: if you look up at an object, the angle your line of sight makes with a horizontal line. Angle of Depression: if you look down at an object, the angle your line of sight makes with a horizontal line Solve a Right Triangle: means to find the measures of all of its sides and angles. You can solve a right triangle if you know either two side lengths of one side length and the measure of one acute angle. Chapter 8 Diagonal: a segment that joins two nonconsecutive vertices Parallelogram: a quadrilateral with both pairs of opposite sides parallel. Rhombus: a parallelogram with four congruent sides. Rectangle: a parallelogram with four right angles. Square: a parallelogram with four congruent sides and four right angles. Trapezoid: a quadrilateral with exactly one pair of parallel sides. Bases: the parallel sides of a trapezoid. Base Angles: a trapezoid has two pairs of these angles. Legs of a Trapezoid: the nonparallel sides of the trapezoid. Isosceles Trapezoid: if the legs of a trapezoid are congruent. Midsegment of a Trapezoid: is the segment that connects the midpoints of its legs. Kite: a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Chapter 9 Image: when a transformation moves or changes a figure in some way to produce a new figure. Preimage: another name for the original figure. Isometry: a transformation that preserves length and angle measure. Vector: is a quantity that has both direction and magnitude, or size. Initial Point: the starting point of the vector. Terminal Point: the ending point of a vector. Component Form: combines the horizontal and vertical components. Matrix: a rectangular arrangement of numbers in rows and columns. Element: each number in a matrix Dimensions: are the numbers of rows and columns. Line of Reflection: the mirror line that reflects and image. Center of Rotation: a rotation is a transformation in which a figure is turned about a fixed point Angle of Rotation: rays drawn from the center of rotation to a point and its image. Glide Reflection: a transformation in which every point P is mapped to a point Pβ, by the following steps: Step 1 β A translation maps P to Pβ. Step 2 β A reflection in a line k parallel to the direction of the translation maps Pβ to Pβ. Composition of Transformation: when two or more transformations are combined to form a single transformation. Line Symmetry: a figure in the plane has this if the figure can be mapped onto itself by a reflection in a line. Line of Symmetry: line of reflection for a figure to map onto itself. Rotational Symmetry: a figure in a plane has this if the figure can be mapped onto itself by a rotation of 180° or less about the center of the figure. Center of Symmetry: the point at which the rotation of the figure happens. Scalar Multiplication: the process of multiplying each element of a matrix by a real number or a scalar. Chapter 10 Circle: the set of all points in a plane that are equidistant from a given point. Center: the middle point that all points of a circle are equidistant from. Radius: a segment whose endpoints are the center and any point on the circle. Chord: is a segment whose endpoints are on a circle. Diameter: a chord that contains the center of the circle. Secant: is a line that intersects a circle in two points. Tangent: a line in a plane of a circle that intersects the circle in exactly one point, the point of tangency. Central Angle: is an angle whose vertex is the center of the circle. Minor Arc: if mοπ΄π΅πΆ is less than 180°, then the points on circle C that lie in the interior of οπ΄π΅πΆ form the minor arc. Μ form the major arc. Major Arc: the points on circle C that do not lie on minor arc π΄π΅ Semicircle: an arc with endpoints that are the endpoints of a diameter. Measure of Minor Arc: the measure of the central angle. Measure of a Major Arc: the difference between 360° and the measure of the related minor arc. Congruent Circles: congruent if they have the same radius. Congruent Arcs: congruent if they have the same measure and they are arcs of the same circle or of congruent circles. Inscribed Angle: an angle whose vertex is on a circle and whose sides contains chords of the circle. Intercepted Arc: the arc that lies in the interior of an inscribed angle and has endpoints on the angle. Inscribed Polygon: a polygon that has all of its vertices lying on a circle. Circumscribed Circle: the circle that contains the vertices. Segments of the Chord: when two chords intersect in the interior of a circle, each chord is divided into two segments. Secant Segment: a segment that contains a chord of a circle, and has exactly one endpoint outside the circle. External Segment: the part of a second segment that is outside the circle. Standard Equation of a Circle: the standard equation of a circle with center (h, k) and radius r is (π₯ β β)2 + (π¦ β π)2 = π 2 Chapter 11 Bases of a Parallelogram: either pair of parallel sides. Height of a Parallelogram: the perpendicular distance between these bases. Height of a Trapezoid: the perpendicular distance between its bases. Circumference: is the distance around the circle. Arc Length: is a portion of the circumference of a circle. Sector of a Circle: the region bounded by two radii of the circle and the intercepted arc. Center of the Polygon: the center of its circumscribed circle. Radius of the Polygon: the radius of the circumscribed circle. Apothem of the Polygon: the distances from the center to any side of the polygon. Central Angle of a Regular Polygon: an angle formed by two radii drawn to consecutive vertices of the polygon. Probability: is a measure of the likelihood that the event will occur. Geometric Probability: a ratio that involved a geometric measure such as length or area. Chapter 12 Polyhedron: a solid that is bounded by polygons. Faces: the planes that enclose a single region of space. Edge: a line segment formed by the intersection of two faces. Vertex: a point where three or more edges meet. Regular Polyhedron: all of its faces are congruent regular polygons. Convex Polyhedron: any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. Platonic Solids: there are five regular polyhedral, regular tetrahedron, cube, regular octahedron, regular dodecahedron, regular icosahedron. Cross Section: the intersection of the plane and the solid. Prism: a polyhedron with two congruent faces that lie in parallel planes. Lateral Faces: the other faces are parallelograms formed by connecting the corresponding vertices of the bases. Lateral Edges: the segments connecting these vertices. Surface Area: the sum of the areas of its faces. Lateral Area: the sum of the areas of its lateral faces. Net: the two-dimensional representation of the faces. Right Prism: each lateral edge is perpendicular to both bases. Oblique Prism: a prism with lateral edges that are not perpendicular to the bases. Cylinder: a solid with congruent circular bases that lie in parallel planes. Right Cylinder: the segment joining the centers of the bases is perpendicular to the bases. Pyramid: a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex. Vertex of the Pyramid: the common vertex between the lateral faces. Regular Pyramid: a regular polygon for a base and the segment joining the verte and the center of the base is perpendicular to the base. Slant Height: the height of a lateral face of the regular pyramid. Cone: has a circular base and a vertex that is not in the same plane as the base. Right Cone: the segment joining the vertex and the center of the base is perpendicular to the base, and the slant height is the distance between the vertex and a point on the base edge. Lateral Surface: consists of all segments that connect the vertex with points on the base edge. Volume: the number of cubic units contained in its interior. Sphere: the set of all points in space equidistant from a given point. Center: the point that all points in space are equidistant from. Radius of a Sphere: a segment from the center to a point on the sphere Chord of a Sphere: a segment whose endpoints are on the sphere. Diameter of a Sphere: a chord that contains the center. Great Circle: if the plane contains the center of the sphere, then the intersection is a great circle Hemispheres: every great circle of a sphere separates the sphere into two congruent halves. Similar Solids: two solids of the same type with equal ratios of corresponding linear measures, such as heights or radii.