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Angles and Lines, Euclidean Geometry 2020, Inductive Reasoning/Deductive Reasoning, 2020 Triangles
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Key Concepts:
Terms in this set (95)
Given
Information provided in the problem
Congruent Supplements Theorem
If angles are supplementary to the same angle then those two angles are congruent
Congruent Complements Theorem
If angles are complementary to the same angle then those two angles are congruent
Perpendicular
Two lines that intersect and form a right angle (90°)
Symbol ⟂
Transversal line
A third line that intersects to others
Parallel Lines
Two lines on a plane that never meet. They are always the same distance apart.
Symbol: ||
Skew Lines
Two nonparallel lines that not are not on same plane that do not intersect
Corresponding Angles
Same side of transversal in similar positions. One point interior and one has to be exterior. They have equal measure.
Alternate Interior Angles
Interior angles on alternative sides of the transversal. All are equal.
Same-Side Interior Angles
Two points that are on the same side of the transversal and are inside of the parallel lines. They are Supplementary
Alternate Exterior Angles
parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Alternate Exterior Angles Converse Theorem
If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.
alternate interior angles converse theorem
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
same side interior angles converse theorem
If two lines are cut by a transversal so the same side interior angles are supplementary, then the lines are parallel.
Parallel Slopes
Same slope, different y-int
perpendicular slope
Perpendicular lines have slopes that are the opposite and the reciprocal of the original slope Normal: 1/2 Perpendicular: -2
Line Segment
Part of a line that consists of two endpoints
Angle
Two rays that have the same initial point
Line
Extends forever in two directions
Parallel
Two lines in the same plane that do not intersect
Collinear
points on the same line
Bisect
to divide into two equal parts
Equiangular
all angles are congruent
Vertical angles
Complementary angles
two angles whose measures have a sum of 90 degrees
Supplementary angles
two angles whose measures have a sum of 180 degrees
Linear pair
Midpoint
A point that divides a segment into two congruent segments
point
An exact location represented by a dot.
Ray
A part of a line, with one endpoint, that continues without end in one direction
Vertex
A point where two or more straight lines meet.
perpendicular lines
Lines that intersect to form right angles
plane
a flat surface that has no thickness and extends forever
arc of a circle
two points on the circle and the continuous part of the circle between the two points
circle
The set of all points in a plane that are the same distance from a given point called the center
conjecture
an opinion or conclusion formed on the basis of incomplete information
Counterexample
a specific case for which the conjecture is false
paragraph proof
A style of proof in which the statements and reasons are presented in paragraph form.
flow proof
Uses arrows to show the flow of the logical argument
two-column proof
a type of proof written as numbered statements and reasons that show the logical order of an argument
Algebra Properties
Addition, Subtraction, Multiplication, Division, Substitution, Reflexive, Symmetric, Transitive, Distributive
Addition and Subtraction Properties
For all numbers a, b, and c, if a=b, then a+c=b+c and a-c=b-c
Multiplication Property of Equality
If a=b, then ac=bc
Substitution Property of Equality
If a=b, then b can be substituted for a in any expression
Symmetric Property of Equality
if a=b, then b=a
Transitive Property
If a=b and b=c, then a=c
Angle Addition Postulate
If P is in the interior of <RST, then m<RSP + m<PST = m<RST
Segment Addition Postulate
If B is between A and C, then AB + BC = AC
inductive reasoning
reasoning based on observations and patterns
conjecture
statement believed to be true based on inductive reasoning
counter example
evidence to proving a conjecture false
conditional statement
statement that can be written in form of 'if p then q'
hypothesis
the part of p of a conditional statement following the word if
conclusion
the part of q of a conditional statement falling the word then
negation
not p
conditional statement
if p then q
converse
if q then p
inverse
if not p then not q
contrapositive
if not q then not p
deductive reasoning
reasoning based on facts, logic, and definitions
Venn Diagram
A diagram that uses circles to display elements of different sets. Overlapping circles show common elements.
truth table
a convenient method for organizing the truth values of statements
Conjuction
a word that connects two or more words or sentences with the word and notated by p^q
disjunction
a compound statement formed by joining two or more statements with the word or; notated by p v q
SSS
Three sides of one triangle are congruent to three sides of another triangle.
SAS
Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
ASA
Two angles and the included side of one triangle are congruent to two triangles and the included side of another triangle.
AAS
Two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle.
HL
The hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle.
Perpendicular Bisector
A line that forms 90 degree angles and cuts a segment in half. Any point on the perpendicular bisector is equidistant to the endpoints of the segment.
Angle Bisector
A ray or line that cuts an angle in half. Any point on the angle bisector is equidistant to the sides of the angle.
Corresponding parts of congruent triangles are congruent (CPCTC)
A theorem stating that if two triangles are congruent, then so are all corresponding parts.
Congruent Sides
Have the exact same length.
Vertex of an Angle
A corner point of an angle. For an angle, the vertex is where the two rays making up the angle meet.
Corresponding Sides
Two sides that are situated the same way in different objects.
Corresponding Angles
Two angles that are situated the same way in different objects.
Third Angles Theorem
If the 2 angles of one triangle are congruent to 2 angles of another triangle, then the third angles are congruent
Isosceles Triangle Theorem
If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent
Converse of the Isosceles Triangle Theorem
If 2 angles of a triangle are congruent, then the sides opposite of the angles are congruent
Congruent
when two parts of a geometric figure have the same shape and size
Corresponding
geometric parts that are in the same position in different figures
Included side
A side that is between two angles in a triangle
In the figure, LM is the included side between angle L and angle M.
Included angle
An angle that is between two sides in a triangle
In the figure, angle N is the included angle between LN and MN.
Angle-Angle-Angle (AAA)
does NOT prove triangles congruent because the sides could be different lengths.
Side-Side-Angle (SSA)
does NOT prove triangles congruent
Congruent Triangles
3 pairs of corresponding angles are congruent and 3 pairs of corresponding sides are congruent.
Equiangular Triangle
A triangle whose angles are all congruent.
Scalene Triangle
A triangle that has no congruent sides.
Isosceles Triangle
A triangle with two congruent sides.
Equilateral Triangle
A triangle that has all three sides congruent.
Triangle Sum Theorem
The sum of the measures in a triangle is 180 degrees.
Exterior Angle Theorem
The sum of the remote interior angles is equal to the measure of the exterior angle.
Hinge Theorem
Longest side is opposite largest angle; same with smallest and middle
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
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