Geometry Unit 3 Vocabulary Angles and Lines Adjacent angles - angles that share a vertex and a ray and no interior points Alternate interior angles – nonadjacent angles formed by two lines and a transversal, with interiors inside the two lines and on different sides of the transversal. Alternate interior angles are congruent. Alternate exterior angles – nonadjacent angles formed by two lines and a transversal, with interiors outside the two lines and on different sides of the transversal. Alternate exterior angles are congruent. 1. Auxiliary line – a dotted line or segment drawn in a diagram to help in solving a problem or proving a concept. There are a series of postulates that are used to support the addition of an auxiliary line to a proof, such as Two points determine one unique line or segment. Each angle has one unique angle bisector. Through a point not on a line (segment), only one line can be drawn parallel to the given line (segment). (Parallel Postulate) Through a point not on a line (segment), only one line can be drawn perpendicular to the given line (segment). Through a point not on a segment, only one line (segment) can be drawn to the midpoint of the given segment. Axiom/postulate – an accepted truth needed to explain undefined terms and to serve as the starting point for proving other statements, called theorems. Complementary Angles - TWO angles the sum of whose measures is 90. Consecutive Angles – two angles that share a common side. Corollary/Theorem – a statement or conjecture that can be proven to be true based on postulates, definitions, or other proven theorems Corresponding Angles –a pair of angles in similar locations with respect to a transversal and its two intersected lines. Corresponding angles are congruent. Definition – a precise description that clearly and uniquely specifies an object or class of objectss Properties of a good definition: •Name the term to be defined •Place the term to be defined in the smallest possible class. •Tell how it is different (sets it apart) from the other members of that class. Exterior angle of a polygon– an angle that forms a linear pair with an angle of a polygon. Interior angle of a polygon– an angle within a polygon. Line segment – a part of a line between two endpoints Linear pairs – two adjacent angles whose noncommon sides are opposite rays forming a straight line. Linear pairs are supplementary. Midpoint – a point that is directly in the middle and bisects a segment (into two equal pieces) Parallel Lines ( || ) – lines in the same plane that do not intersect; they remain an equal distance apart. (Parallel lines have equal slopes) Parallel lines cut by a transversal - If lines are parallel then the following pairs of angles are congruent. Postulate/axiom – an accepted truth needed to explain undefined terms and to serve as the starting point for proving other statements, called theorems. Proof – a sequence of justified conclusions, leading from what is given or known to a final conclusion Same-side interior angles – angles formed by two lines and a transversal, with interiors inside the two lines and on the same side of the transversal. Same-side interior angles are supplementary. Sum of Interior Angles of a Triangle Theorem – the sum of the interior angles of a triangle is 180o Theorem/Corollary – a statement or conjecture that can be proven to be true based on postulates, definitions, or other proven theorems Transversal – a line that intersects two or more lines Triangle – a polygon with 3 sides and 3 angles. The sum of the 3 interior angle measures of any triangle is 180⁰. Triangle Exterior Angle Theorem – the measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. Undefined Terms– the “three undefined terms of geometry” are point, line, and plane. They have no formal definition, but they do have descriptions. Point – an exact location with no thickness Line – a set of points extending in both directions containing the shortest path between any two points in it; it has no width or height. Plane – a flat surface without any boundaries or edges Vertical Angles – two non-adjacent congruent angles formed by two intersecting lines. Postulate/Axioms/Theorems/Corollaries Angle Measure Postulates Unique Measure Assumption: Every angle has a unique measure from 0o to 180o Two Sides of a Line Assumption: Given any ray VA and any number x between 0o to 180o, there are unique rays VB and VC such that line segment BC intersects line VA and m< BVA = m< CVA= xo. Zero Angle Assumption: If rays VA and VB are opposite rays, then m<AVB=180o Angle Addition Property: : If ray VC (except for point V) is in the interior of <AVB, then m<AVC + m<CVB = m<AVB Point-Line-Plane Postulates Unique Line Assumption - Through any two points, there is exactly one line. Dimension Assumption - Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane. Number Line Assumption - Every line is a set of points that can be put into a 1-1 correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1. Distance Assumption - On a number line, there is a unique distance between two points. Postulates of Equality Reflexive Property of =: a=a Symmetric Property of =: If a=b, then b=a Transitive Property of =: If a = b and b=c, then a=c Postulates of Equality & Operations Addition Property of Equality: If a=b, then a+c=b+c. Multiplication Property of Equality: If a=b, then ac=bc. Substitution Property of Equality: If a=b, then a can be substituted in any expression. Postulates of Inequality & Operations Addition Property of Inequality: If a<b, then a+c<b+c. Multiplication Property of Inequality: If a<b and c>0, then ac<bc. If a<b and c<0, then ac>bc. Equation to Inequality Property: If a and b are positive numbers and a+b=c, then c>a and c>b. Transitive Property of Inequality: If a and b are positive numbers and a+b=c, then c>a and c>b. Postulates of Operations Commutative Property of Addition: a+b=b+a Commutative Property of Multiplication: ab=ba Distributive Property: a(b+c)=ab+ac

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# Unit 3