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Geometry 3rd GP Notes 032212 Pointers 1st and 2nd Term Undefined and basic terms Angles and angle pairs Parallel lines and theorems 3rd Term Areas of polygonal regions Circles Similarities Solids I. Undefined Terms Half Line set of points which is the union of all the points of a line on one side of a given point excluding the given point Plane flat surface usually represented by four sided figures has infinite length and width but no thickness named by 3 non collinear points or a capital letter on the corner Point Line Line Segment Ray Opposite Rays Half Lines no formal definition can only be represented or described can also be dealt with through their properties which are taken up as postulates Point represented by a dot named by a capital letter points contained on the same line Non collinear Points points not contained on the same line Coplanar Points points contained on the same plane Non coplanar Points points not contained on the same plane Line set of points extending infinitely in both directions has length but no width nor thickness arrowhead named using 2 points on each end or a small letter in the middle Line Segment set of points in between two points named by its endpoints definite Ray part of a line with one endpoint and extends infinitely in the other direction named by two points, name starting at end points Opposite Rays two rays with a common endpoint contained on the same line but going in opposite directions cpablo Plane A II. Basic Terms Postulates assumed to be true also known as an Axiom accepted without proof Theorems further need to be proved Space set of all points Collinear points are located on the same line Non Collinear points are not located on the same line Coplanar points are located on the same plane Non Coplanar points are not located on the same plane Congruent Segments segments with the same length Midpoint of a Segment divides the segment into two congruent parts also the bisector of a segment Betweenness 1|P a g e not all betweenness are midpoints but all midpoints are betweenness Bisector midpoint of a segment and may be a line, half line, ray, or segment passing through the midpoint all midpoints are bisectors but not all bisectors are midpoints III. Angles and Angle Pairs Angle union of two non collinear rays with a common endpoint Angle Bisector- a ray bisects an angle if it divides the angle into two congruent angles Congruent Angles- angles whose measurements are equal Perpendicular lines- forms 4 90 degree angles Perpendicular bisector- cuts into 2 congruent 90 degree parts Angle bisector postulate- one angle bisector in every angle Angle addition postulate- add two angles to make a bigger angle Types of Angles Acute Angle : measures less than 90 Right Angle: measure is equal to 90 Obtuse: measure is more than 90 but less than 180 Angle Pairs Complementary Angles two angles whose measures add up to 90 Supplementary Angles two angles whose measures add up to 180 Adjacent Angles two angles with a common side but do not have common interior points Vertical Angles two nonadjacent angles formed by two intersecting lines Linear Pair two adjacent angles whose non common sides also form a pair of opposite rays IV. Parallel Lines and Planes (not sure if we need the other theorems but I added everything) Parallel Lines lines which are coplanar and do not intersect cpablo Skew Lines lines which are non coplanar and do not intersect. Parallel Lines Theorem “Two parallel lines lie in exactly one plane.” Transversal a line which intersects two or more coplanar lines at two or more distinct points. Same-Side Interior Angles a pair of nonadjacent interior angles on the same side of the transversal Alternate Interior Angles a pair of nonadjacent interior angles on opposite sides of the transversal Alternate Exterior Angles a pair of nonadjacent exterior angles on opposite sides of the transversal Corresponding Angles a pair of nonadjacent angles, one interior, one exterior, on the same side of the transversal PCA Postulate “If two parallel lines are cut by a transversal, then the corresponding angles are congruent.” PAI Theorem “If two parallel lines are cut by a transversal, then the alternate-interior angles are congruent.” PAE Theorem “If two parallel lines are cut by a transversal, then the alternate-exterior angles are congruent.” PIS Theorem “If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary.” CAP Postulate “If two lines are cut by a transversal and a pair of corresponding angles are congruent, then the lines are parallel.” AIP Theorem “If two lines are cut by a transversal and a pair of alternate-interior angles are congruent, then the lines are parallel.” 2|P a g e AEP Theorem “If two lines are cut by a transversal and a pair of alternate-exterior angles are congruent, then the lines are parallel.” ISP Theorem “If two lines are cut by a transversal and a pair of same-side interior angles are supplementary, then the lines are parallel.” Parallel Postulate “Through an external point, there is exactly one line parallel to the given line.” Transitivity of Parallelism Theorem (TPT) or Transitivity of Relation Parallel (TRP) “In a plane, if two lines are parallel to the same line, they are parallel to each other.” Theorem 9.9 “In a plane, two lines perpendicular to the same line are parallel.” it is convex if each line that contains a side of the polygon contains not a single point in the interior of the polygon it is regular if it is convex, equilateral and equiangular it is inscribed in a circle if the vertices of the polygon lie on the circle it is circumscribed about a circle if each side of the polygon intersects the circle at only one point Quadrilateral four sided polygon Parallelogram quadrilateral in which both pairs of opposite sides are parallel Trapezoid quadrilateral with exactly one pair of opposite sides parallel Kite quadrilateral in which one diagonal is perpendicular bisector of the other diagonal Theorem 9.10 “In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.” Rectangle parallelogram with four right angles Parallel Planes planes that do not intersect Rhombus parallelogram with four congruent sides Theorem 9.11 “If two planes are perpendicular to the same line, then the planes are parallel.” Square rectangle with four congruent sides Theorem 9.12 “If two parallel lines are cut by a third plane, then the lines of intersection are parallel.” V. Polygons and Polygonal Regions (not sure if we still need the definitions but I added everything) Polygon union of three or more coplanar segments such that each segment intersects exactly two other segments (one at each endpoint) and no two intersecting segments are collinear. parts: sides, vertices, angle; consecutive sides, opposite sides, diagonals three sets of points: polygon, interior, exterior cpablo the Properties of a Parallelogram opposites are congruent consecutive angles are supplementary each diagonal separates the parallelogram into 2 congruent triangles diagonals bisect each other Midsegment segment joining midpoints of each side of a triangle the length of the midsegment is always half of the base Properties of a special parallelogram diagonals of a rectangle are congruent if a parallelogram has one right angle then it is a rectangle diagonals of a rhombus bisect opposite angles 3|P a g e Alternation 1/3=2/6 Inversion 6/3=2/1 Addition 1+2/2=3+6/6 Subtraction 1-2/2=3-6/6 Summation 1/2=3/6=1+2/3+6 Properties of an Isosceles Trapezoid base angles are congruent diagonals are congruent trapezoid median= b1+b2/2 Sum of Interior and Exterior angles of Polygons Sn (Sum of Interior angles)= (n-2)180 Se (Measurement of each Exterior angle)= 360/n Diagonal= n(n-3)/2 Area of Polygons Area of a Square: A = s2 Area of a Rectangle: A = lw Area of a Parallelogram: A = bh Area of a Triangle: A = ½bh Area of a Right Triangle: A = ½l1l2 Area of a Equilateral Triangle: A = (s2√3)/4 Area of a Rhombus: A = ½d1d2 Area of a Trapezoid: A = ½h(b1 + b2) Area of a Regular Polygon: A = ½ap VIII. Similar Polygons and Triangles Definition Similar: have congruent angles and proportional sides Congruent: have congruent angles and sides, exactly the same size and shape Postulates to determine Similarity AA SAS SSS VI. Special Right Triangles 45 60 IX. Basic Proportionality x sqrt 2 2x x x 30 x sqrt 3 30 – 60 – 90 Triangle Definition 2 4 45 Definition Ratio 1 : 2 or 1 / 2 Relationship of 2 numbers Proportion 1/2=1/4 Equality of 2 numbers Properties 1/ 2 = 3/ 6 Means and Extremes 1x6=2x3 cpablo 14 x Theorems A 45 – 45 – 90 Triangle B Pythagorean Theorem- a2 + b2 = c2 VII. Similarity and Proportion 7 C D E Basic Proportionality Theorem (BPT) If segment BE // segment CD then AB / BC = AE / ED ; AC / AB = AD / AE Converse of the Basic Proportionality Theorem (CBPT) If AB / BC – AE / ED then segment BC // segment CD Geometric Mean x, y, and z are all real numbers x/y=y/z y2 = xz y = sqrt xz 4|P a g e Theorem 13. 7 Altitude- line where 2 lines intersect Tangent circles: two coplanar circles that are tangent to the same line at the same point Internally tangent circles: two coplanar circles whose centers are on the same side of their common tangent Externally tangent circles: coplanar circles whose centers are on opposite sides of their common tangent Area of Similar Triangles 5 2 1 2 2 / 5 = (2 / 5)2 = 4 / 25 X. Parts and Theorems of Circles Refer to Chapter 15 for the figures Basic Terms Circle: set of all points in a plane at a certain distance from a center point Radius: segment whose endpoints are the center of the circle and a point on the circle Diameter: segment containing the center and whose endpoints are points on the circle Interior: set of points whose distance is less than the radius Exterior: set of points whose distance is greater than the radius Chord: segment whose endpoints are points on the circle Secant: line that intersects a circle in exactly two points Tangent: line in the plane of the circle that intersects the circle at exactly one point Ray: segment that is a subset of a tangent line and contains the point of tangency Relations of Circles Congruent circles: circles with congruent radii Concentric circles: two or more circles with the same center Common tangent: line that is tangent to each of two coplanar circles Common external tangent: does not intersect the segment joining the centers of the circle Common internal tangent: intersects the segment joining the centers of the circle cpablo Tangent, Secants, and Chords Theorem 15.1: “A line perpendicular to a radius at its outer end is tangent to the circle.” Theorem 15.2: “Every tangent to a circle is perpendicular to the radius drawn to the point of tangency.” Theorem 15.3: The Two-Tangent Theorem: “The two tangent segments to a circle from an external point are congruent and determine congruent angles with the segment from the external point to the center.” Theorem 15.4: “The line from the center of a circle perpendicular to a chord, bisects the chord.” Theorem 15.5: “The line containing the center of a circle and the midpoint of a chord that is not a diameter, is perpendicular to the chord.” Theorem 15.6: “In the plane of the circle, the perpendicular bisector of a chord passes through the center.” Corollary 15.6-1: “No three points of a circle are collinear.” Theorem 15.7: “In the same or in congruent circles, chords equidistant from the center are congruent.” Theorem 15.8: “In the same or in congruent circles, any two congruent chords are equidistant from the center.” Theorem 15.9: “The graph of the equation (x-h)2 + (y-k)2 = r2 is the circle with center (h, k) and radius r.” Theorem 15.10: “Every circle is the graph of the equation of the form: x2 + y2 + Ax + By + C = 0.” Theorem 15.11: “The graph of the equation x2 + y2 + Ax + By + C = 0 is a circle (+), a point (0), or an empty set (-).” Arcs and Angles Related to a Circle Arc: subset of a circle with two endpoints Central angle: angle in the plane of the circle whose vertex is the center of the circle Minor arc: set of points on the circle that lie on a central angle or in the interior of the central angle 5|P a g e Major arc: set of points on the circle that lie on a central angle or in the exterior of the central angle Semicircle: union of the endpoints of a diameter and the points of the circle on one side of the diameter Degree measurement of a circle: 360 Degree measurement of a semi circle: 180 Degree measurement of a minor arc: measure of its central angle Degree measurement of a major arc: 360 – minor arc Power Theorems Postulate 33: Arc Addition Postulate: “If Q is on arc RS then measurement of arc SQ + m arc QR = m arc SQR.” Congruent arcs: arcs that have the same measure and lie on the same or on congruent circles Angle inscribed: angle whose sides contain the endpoints of an arc and whose vertex is a point on the arc other than the endpoints Intercepted arc: arc whose endpoints lie on difference sides of an angle and whose other points lie in the interior of the angle Theorem 15.12: Inscribed Angle Theorem: “the measure of an inscribed angle is one-half the measure of its intercepted arc.” Corollary 15.12-1: “Two inscribed angles that intercept the same arc or congruent arcs are congruent.” Corollary 15.12-2: “An angle inscribed in a semicircle is a right angle.” Corollary 15.12-3: “The opposite angles of an inscribed quadrilateral are supplementary.” Theorem 15.13: “The measure of an angle formed by two secants which intersect in the interior of a circle is one –half the sum of the measures of the arcs intercepted by the angle and its vertical angle.” Theorem 15.14: “The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.” Theorem 15.15: “The measure of an angle formed by a tangent and a secant that intersect at the point of tangency is one-half the measure of the intercepted arc.” Theorem 15.16: “The measure of an angle formed by a secant and a tangent that intersect in the exterior of a circle is one-half the difference of the measure s of the intercepted arcs.” Theorem 15.17: “The measure of an angle formed by two intersecting tangents is one-half the difference of the measures of the intercepted arcs. cpablo Theorem 15.18: The Two-Chord Power Theorem: “If two chords intersect in a circle, then the product of the lengths of the segments on one chord is equal to the product of the lengths of the segments on the other.” If a segment intersects a circle in two points, and exactly one of these is an endpoint of the segment, then the segment is called a secant segment to the circle. The part of the secant segment in the exterior of the circle is called its external secant segment. Theorem 15.19: The Two-Secant Power Theorem: “If two secants intersect at a point in the exterior of the circle, the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.” Theorem 15.20: The Tangent-Secant Power Theorem: “If a tangent and secant intersect at a point in the exterior of the circle, the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.” XI. Area and Circumference of Circles and Arcs Area and Circumference of a Circle Limit: fixed number which a sequence of numbers approaches Circumference: limit of the perimeters of the inscribed regular polygons as the number of sides increases Area: limit of the ares of the inscribed polygons as the number of sides increases Ratio of the circumference of a circle to the diameter is denoted by , = c / d Inscribed and Circumscribed “A circle is inscribed in a polygon when the sides of the polygon are tangent to the circle.” “A circle is circumscribed about a polygon when the vertices of the polygon lie on the circle.” Sectors and Segments of a Circle Sector: region bounded by two radii and the intercepted minor or major arc Theorem 15.21: “In a circle of radius r, the ratio of the length L of an arc to the circumference C of the circle is the same as the ratio of the arc measure m to 360.” L / C = m / 360 or L = m / 180 r Asector= m / 360 r2 6|P a g e Segment: Region bounded by a chord and the intercepted minor or major arc Sphere: set of points in space that are at a given distance, called the radius, from a given point, called the center XII. Prisms Solids and Polyhedrons Solid: three-dimensional figure bounded by flat surfaces, curved surfaces, or both flat and curved surfaces Polyhedron: union of four or more noncoplanar polygonal regions which encloses a part of space Enclose space: interior of the polyhedron Solid Polyhedron: union of a polyhedron and its interior Area and Volume of Prisms Prism: polyhedron with two congruent faces contained in parallel planes and its other faces are parallelogram regions The bases are congruent and lie on parallel planes. The lateral faces are parallelogram regions. The lateral edges are parallel and congruent. Regular Prism: right prism whose bases are regular polygonal regions Parallelepiped: prism whose bases are parallelogram regions Rectangular parallelepiped: right rectangular prism Cube: rectangular parallelepiped all of whose edges are congruent Postulate 34: Volume Postulate: “To every solid region there corresponds a unique positive real number called the volume of the solid region.” Postulate 35: Congruent Solids Postulate: “If two solids are congruent, then they have exactly the same volume.” Postulate 36: Volume Addition Postulate: “If a solid region is separated into nonoverlapping regions, then the sum of the volumes of these regions equals the volume of the given region.” Cylinder: three-dimensional figure bounded by a curved surface with two congruent circular bases contained in parallel planes Pyramid: polyhedron formed by a polygonal region in a plane, a point not on the plane, and the triangular regions formed by joining the vertices of the polygonal region by the point Cone: three-dimensional figure bounded by a curved surface with a vertex not on the plane and a circular region base cpablo Formulas for Prisms Prism d= sqrt l2 + w2 + h2 sum of all lateral faces TA = 2B + LA V= bh Cylinder LA = 2rh TA = 2r(h + r) V= r2h Pyramid LA= 1 / 2 (ps) TA= LA + B V= 1 / 3 (bh) Cone LA = rs TA = LA + B V = 1 / 3 bh Sphere SA = 4r2 VA = 4 / 3 r3 7|P a g e