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Foundations of Euclidean Geometry - Unit 1 by

Euclidean Geometry

a statement that has been proven based on previous theorems, postul­ates, or axioms
points that lie on the same line
deductive reasoning
the process of utilizing facts, proper­ties, defini­tions, and theorems to form a logical argument
contained within the same plane
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

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

Measuring Length and Angles

a point on a line segment that is equidi­stant from the two endpoints
tool used to measure an angle in degrees
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

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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