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8th Grade Geometry Terms, Postulates and Theorems UNDEFINED TERMS: Point – no measurable length, no measurable width, no measurable depth; named with one capital letter. Line – measurable length, no measurable width, no measurable depth; named with 2 capital letters (representing the names of 2 points on the line) and a symbol of a line above the letters, or with one cursive small letter. Example: AB Plane – measurable length, measurable width, no measurable depth; named with 3 capital letters (representing the names of 3 points not on the same line) and the word plane in front. Example: Plane ABC DEFINITIONS: Collinear – points, segments or rays on the same line. Coplanar – points, lines, segments, or rays on the same plane. Line segment or segment – that portion of a line with 2 distinct endpoints; Named with 2 capital letters (the names of the endpoints) and a symbol of a segment above the letters. Example: AB Ray – that portion of a line with 1 endpoint; named with 2 capital letters (the first being the endpoint of the ray, and the second being a point on the ray) and a symbol of a ray above the letters. Example: AB Opposite Rays – Two rays that share the same endpoint and extend in opposite directions to form a line. Length of a Segment – a real number representing the length of a segment – written with 2 capital letters (the names of the endpoints). Example: AB = 5 the length of segment AB is 5 units. Midpoint – a point that bisects a segment. If A is the midpoint of CD , then CA = AD. Angle – the figure created when 2 rays share the same endpoint; Named with 3 letters (the first letter is the name of a point on one side, the second letter is the name of the shared endpoint, and the third letter is the name of a point on the other side) and an angle sign in front of the letters or a number with the angle sign in front of the number. Example: ABC Sides of an angle – the two rays that form the angle. Vertex of an angle – the shared endpoint of the 2 rays that form the angle. Interior of an angle – the points that are between the sides of the angle. Exterior of an angle – the points that are not in the interior or on the angle. Measure of an angle – a real number from 0 to 180, representing the measure of the angle using a protractor. Named with a non-capital letter m in front of the name of the angle. Example: m ABC = 80°, the measure of angle ABC is 80 degrees. Adjacent Angles – Two angles that share a side, and share a vertex, but do not share any interior points. Acute Angle – an angle that measures less than 90°. Right Angle – an angle that measures 90°. Obtuse Angle – an angle that measures more than 90° but less than 180°. Straight Angle – an angle that measures 180°. Vertical Angles – two non-adjacent angles formed when 2 lines intersect. Complementary Angles – two angles whose measures add up to 90°. Supplementary Angles - two angles whose measures add up to 180°. Bisector – to cut into 2 congruent parts. Linear Pair – 2 adjacent angles whose non-shared sides are opposite rays. Parallel Lines – coplanar lines that do not intersect. (Symbol //) Skew Lines – non-coplanar lines that do not intersect. Perpendicular Lines – two coplanar lines that meet to form a right angle. (Symbol ) Congruent – to have the same measure. Parallel Planes – two planes that do not intersect. Transversal – a line that intersects 2 or more lines. Corresponding Angles – the angles formed when 2 lines are cut by a transversal and correspond to the same position at the points of intersection. Alternate Interior Angles – the angles formed when 2 lines are cut by a transversal and are between the lines, on opposite sides of the transversal and at different points of intersection. Alternate Exterior Angles - the angles formed when 2 lines are cut by a transversal and are outside the lines, on opposite sides of the transversal and at different points of intersection. Consecutive Interior Angles - the angles formed when 2 lines are cut by a transversal and are between the lines, on the same side of the transversal and at different points of intersection. Triangle – A closed figure with 3 angles and 3 sides; Named with 3 capital letters representing the names of the vertices of the triangle and a symbol of a triangle in front of the letters. Example: ABC Scalene Triangle – a triangle with 3 non-congruent sides. Isosceles Triangle – a triangle with 2 congruent sides. Equilateral Triangle – a triangle with 3 congruent sides. Acute Triangle – a triangle with 3 acute angles. Right Triangle – a triangle with 1 right angle. Obtuse Triangle – a triangle with 1 obtuse angle. Equiangular Triangle – a triangle with 3 congruent angles. PROPERTIES OF EQUALITY AND CONGRUENCE: Reflexive Property of Equality: a = a Symmetric Property of Equality: If a = b, then b = a. Transitive Property of Equality: If a = b and b = c, then a = c. Substitution Property of Equality: If a = b and a + c = d, then b + c = d. Addition Property of Equality: If a = b, then a + c = b + c. Subtraction Property of Equality: If a = b, then a - c = b - c. Multiplication Property of Equality: If a = b, then a · c = b · c. Division Property of Equality: If a = b, then a b = . c c Reflexive Property of Congruence: AB AB Symmetric Property of Congruence: If AB CD , then CD . AB Transitive Property of Congruence: If AB CD and CD XY , then AB . XY POSTULATES AND THEOREMS: If point A is between points C and D, then AC + AD = AD. All right angles are congruent. If 2 angles are supplementary to the same or congruent angles, then they are congruent to each other. If 2 angles are complementary to the same or congruent angles, then they are congruent to each other. If 2 angles form a linear pair then they are supplementary. (Linear Pair Postulate). Vertical angles are congruent. If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent. If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. If 2 parallel lines are cut by a transversal, then the alternate exterior angles are congruent. If 2 parallel lines are cut by a transversal, then the consecutive interior angles are supplementary. If 2 lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If 2 sides of 2 adjacent acute angles are perpendicular, then the angles are complementary. If 2 lines are perpendicular, then they intersect to form 4 right angles. In a coordinate plane, if 2 lines are parallel, then their slopes are equal. Any 2 vertical lines are parallel. In a coordinate plane, if 2 lines are perpendicular, then the product of their slopes is -1. Vertical and horizontal lines are perpendicular. If 3 parallel lines intersect 2 transversals, then they divide the transversals proportionally. Pythagorean Theorem The sum of the angles of a triangle is 180°.