Geometry Reasons for Proofs Chapter 1
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1 Geometry Reasons for Proofs Chapter 1 Lesson 1.1 Defined Terms: Undefined Terms: Point: Line: Plane: Space: Postulate 1: Postulate : terms that are explained using undefined and/or other defined terms terms that can only be explained using examples and descriptions (Points, Lines, and Planes) a specific location. A point has no dimensions, no length, width, or depth a collection of points that extend in opposite directions forever, at a constant rate of change. A line has no thickness or width a flat surface made up of points that extend infinitely in all directions a boundless, three dimensional set of all points Through any two points there is exactly one line. Through any three points not on the same line there is exactly one plane. Collinear: If two points are collinear, Iff then the points lie on the same line. Coplanar: If figures are coplanar, Iff then the figures lie in the same plane. Non collinear: If two points are non collinear, Iff then the points do not lie on the same line. Non coplanar: If figures are non coplanar, Iff then the figures do not lie in the same plane. Intersection: the set of all points in common between two or more geometric figures Lesson 1. Line Segment: Betweenness of Points: a portion of a line that is defined by two endpoints For any two real numbers a and b, there is a real number n that is between a and b such that a < n < b.
2 Between: If point B is between point A and point C, Iff then points A, B, and C are collinear. (and) then AB + BC = AC. Segment Addition If point Q is between point P and R, Iff Postulate then PQ + QR = PR. Congruent Segments: If two segments are congruent, Iff then the two segments have the same measure. Construction: Locus: geometric figures that are drawn with a compass and a straight edge a set of points that satisfy a particular condition Lesson 1.3 Distance (number line): The distance between two points is the absolute value of the difference between their coordinates. PQ x x1 x1 x Distance Formula: (coordinate plane) The distance between two point is d ( x x ) ( y y ) 1 1 Midpoint (number line): If AB has endpoints at x 1 and x on a number line, then the midpoint M x1 x of AB has coordinate M. Midpoint: If AB has endpoints at (x 1, y 1 ) and (x, y ) on a number line, then the (coordinate plane) 1 1 midpoint M of AB has coordinate x, M y. Midpoint: Midpoint Theorem: If M is the midpoint of PQ, then PM = MQ. If M is the midpoint of AB, then AM MB. Segment Bisector: If a segment, line or plane is a segment bisector, Iff then it intersects the segment at its midpoint.
3 Lesson 1.4 Ray: Opposite Ray: Angle: Sides: Vertex: Interior of an Angle: Exterior to the Angle: Degrees: a part of a line that has one endpoint and extends indefinitely in one direction two rays that share a common endpoint and extend in opposite directions (also called a straight angle) formed by joining any two noncollinear rays with a common endpoint one of the rays that forms an angle the common endpoint in the angle a point is in the interior of an angle if it does not lie on the angle itself and it lies on a segment with endpoints that are on the sides of the angle. a point is in the exterior of an angle if it is neither on the angle nor in the interior of the angle A unit of measure used in measuring angles and arcs. There are 360 o in a circle. Right Angle: If an angle is a right angle, Iff then its measure is 90 o. Acute Angle: If an angle is an acute angle, Iff then its measure is between 0 o and 90 o. Obtuse Angle: If an angle is an obtuse angle, Iff then it s measure is between 90 o and 180 o. Congruent Angles: If two angles are congruent angles, Iff then the angles have the same measure. Angle Bisector: If a segment, line or ray is an angle bisector, Iff then it divides the angle into two congruent angles.
4 Lesson 1.5 Adjacent Angles: two angles that lie in the same plane and have a common vertex and a common side, but no common interior points Angle Addition If R is in the interior of PQS, Iff Postulate then m PQR + m RQS = m PQS. Linear Pair: If two angles are a linear pair, Iff then they are adjacent angles. (and) then their non common sides are opposite rays. Supplemental Theorem Vertical Angles: Vertical Angles Theorem If two angles are linear pairs, then the two angles are supplementary. two angles that are non adjacent angles that are formed by interesting lines If two angles are vertical angles, then the two angles are congruent. Complementary Angles: If two angles are complementary angles, Iff then the sum of the measures of the two angles is 90 o. Supplementary Angles: If two angles are supplementary angles, Iff then the sum of the measures of the two angles is 180 o. Perpendicular Lines: If two intersecting lines are perpendicular, Iff then they form four right angles. Lesson 1.6 Polygon: Vertex of the Polygon: Concave: Convex: a closed figure formed by a finite number of coplanar segments the common endpoints for the adjacent segments that makeup the polygon a polygon for which there is a line containing a side of the polygon that also contains a point in the interior of the polygon a polygon for which there is no line that contains both a side of the polygon and a point in the interior of the polygon
5 n gon: a polygon with n number of sides Number of Sides Polygon Number of Sides Polygon 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 11 Hendecagon 7 Heptagon 1 Dodecagon Equilateral Polygon: If a polygon is equilateral, Iff then the polygon has all congruent sides. Equiangular Polygon: If a polygon is equiangular, Iff then the polygon has all congruent angles. Regular Polygon: If a polygon is a regular polygon, Iff then the polygon is convex and all of the sides are congruent, and all of the angles are congruent. Perimeter: the sum of the lengths of the sides of a polygon Circumference: the distance around a circle, C r Area: the number of square units needed to cover a surface Figure Perimeter/ Circumference Area Triangle P = b + c + d 1 A bh Square P = 4s A = s Rectangle P = L + W A = bh Circle C r A r
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