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 |
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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 |
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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 |
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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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