Cheatography

# Foundations of Euclidean Geometry - Unit 1 by anjuscha

### Euclidean Geometry

 theorem a statement that has been proven based on previous theorems, postul­ates, or axioms collinear points that lie on the same line deductive reasoning the process of utilizing facts, proper­ties, defini­tions, and theorems to form a logical argument coplanar contained within the same plane postulate a statement accepted without proof; also known as an axiom

 Euclidean geometry is comprised of figures and diagrams that can all be constr­ucted using just a straig­htedge and compass.

### Point, line, plane

 Point Line Plane No dimensions One-di­men­sional set of infinite points Two-di­men­sional set of all points Location on coordinate plane designated by an ordered pair (x/y) Has no beginning or an end Flat or level surface Identified with a single capital letter Identified with a lowercase italicized letter or two capital letters repres­enting two points on the line Identified with a capital italicized letter

### Defining terms

 line segment a part of a line that has two endpoints and a specific length ray part of a line that has one endpoint and extends indefi­nitely in one direction circle the set of all points in a plane that are a given distance away from a given point called the center angle a figure formed by two rays that share a common endpoint parallel lines lines that lie in the same plane and do not intersect perpen­dic­ula­rlines lines that intersect to form right, or 90-degree, angles

### Measuring Length and Angles

 midpoint a point on a line segment that is equidi­stant from the two endpoints protractor tool used to measure an angle in degrees bisect to divide into two congruent parts congruent segments two line segments that have the same length Undefined terms: Point: Points are locations in space. Line: Lines are infinite in two different direct­ions. Defined terms: Line segment: A line segment has two endpoints. Ray: Rays have one endpoint. Angle: An angle is formed by two rays with a common endpoint. adjacent angles two angles within the same plane that share a common side and vertex, but do not share any common interior points congruent angles two angles that have the same measure obtuse angle an angle measuring greater than 90 degrees, but less than 180 degrees straight angle an angle whose measure is exactly 180 degrees acute angle an angle measuring between 0 and 90 degrees right angle an angle whose measure is exactly 90 degrees

### Intro to proof

 conjecture a statement thought to be true but not yet proved true or false deductive reasoning the process of utilizing facts, proper­ties, defini­tions, and theorems to form a logical argument reflexive property the property that states that for any real number 𝑥, 𝑥 = 𝑥; or that a figure and its parts (e.g., sides, angles, triangles, etc.) are congruent to themselves substi­tution property the property stating that if two values are equal, then they are interc­han­geable in an equation; or if two figures are congruent, then they are interc­han­geable in a statement symmetric property thepro­per­tyt­hat­sta­tes­tha­tth­ele­fta­ndright sides of an equation or congruence statement are interc­han­geable Proofs involve: given inform­ation, in words or a diagram, a statement to be proven, an argument using deductive reasoning and justif­ication of steps in a logical order. A conclusion

### Linear Pairs and Vertical Angles

 linear pair 2 adjace­nta­ngles whose noncommon sides are opposite rays vertical angles opposite angles formed by two inters­ecting lines

### Comple­mentary and Supple­mentary Angles

 opposite rays ays that are collinear and have the same endpoint but run infinitely in opposite directions supple­mentary angles two angles whose measures have a sum of 180 degrees comple­mentary angles angles are two angles whose measures have a sum of 90 degrees

### Example finding angle

 Find the following angle measures. 𝑚∠1 = ? 𝑚∠1 + 70° = 90° 𝑚∠1 = 90° − 70 𝑚∠1 = 20

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