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Unit 1 -Tools of Geometry
Point: A location in space
Line: A series of points that extends in 2 opposite directions
Plane: A flat Surface that has no thickness
Collinear Points: Points that are on the same line
Coplanar: Points and lines on the same plane
Line Segment: A part of a line consisting of two endpoints and all the points between them
Ray: A part of a line consisting of one endpoint and all the points on one side
Parallel Lines: Lines in the same plane that do NOT intersect
Skew Lines: Lines that are NON COPLANAR and do not intersect
Angle Addition Postulate
Segment Addition Postulate
Mid Point Postulate
Vertical Angles:
Complimentary angles: angles that Add up to 90 degrees. (angles that form right angles)
Supplementary Angles: Angles that Add up to 180 degrees. (Angles that are on a line)
Adjacent Angles: 2 coplanar angles that share a common side and vertex
5 fundamental Loci
Locus: A set of points that satisfy a given condition
1) A locus of points equidistant from a given point = a circle
Ex: A locus of points 2 units from a point P
2) A locus of points equidistant from a line segment = a track
Ex: The locus of points 5 units from segment
3) A locus of points equidistant from a line = 2 lines
Ex: 3 units from line m
4) A locus of points equidistant from 2 lines(parallel and intersecting) = one line(parallel) or intersecting
lines (intersecting)
Ex: Equidistant from lines l and m(parallel)
5) A locus of points equidistant from 2 points = one line between them
Ex: equidistant from points A and B
l and m (intersecting)
Unit 2 – Reasoning and Proof
Conditional: “if…-then…” statement
Hypothesis: “if” part of the statement, Conclusion – “then” part of the statement
Converse: Switches the hypothesis and conclusion
Inverse: Negates the hypothesis and conclusion
Contrapositive: both switches and negates the hypothesis and conclusion (This is logically equivalent to
the original conditional statement)
Biconditional: “if and only if” statement
Ex: If you are reading this, then you are studying
Converse: If you are studying, then you are reading this
Inverse: If you are not reading this, then you are not studying
Contrapositive: If you are not studying, you are not reading this
Law of Detachment: If the conditional statement is true, and the hypothesis is true, then the statement
is true.
Syllogism Properties of Equality
Addition property:
If a=b then a+c = b+c
Subtraction Property: If a=b then a-c = b-c
Multiplication Property: If a=b then ac=bc
Division Property:
If a=b then a/c=b/c
Reflexive Property:
Symmetric Property: If a=b, then b=a
Transitive Property:
If a=b and b=c, then a=c
Substitution property: If a=b, then “b” can replace “a”
Distributive Property: a(b+c) = ab+bc
Properties of Congruence
Reflexive Property:
Symmetric Property:
Transitive Property:
Unit 3 – Parallel and Perpendicular Lines
Transversal: Line t
Alternate Interior Angles:
Angles are equal
Same Side Interior Angles:
Angles add to 1800
Corresponding Angles:
Angles are equal
Converse Statements:
If the above angles are true, then you have line l parallel to line m.
Triangles and Polygons
Types of Triangles
Equiangular/Equilateral: All sides and angles are congruent
Acute: All angles are less than 900
Right: A triangle with one 900 angle
Obtuse: A triangle with one angle that is bigger than 900
Isosceles: A triangle with 2 equal sides (and two equal angles)
Scalene: A triangle with no equal sides
Triangle Sum Theorem: The sum of the measures of the angles in a triangle is 1800.
Exterior Angle Theorem:
Polygon Interior Angles Theorem: The sum of the measures of the angles inside a polygon is given as
(n-2)180 where “n” is the number of sides of the polygon
Polygon Exterior Angles Theorem: The sum of the degrees in the exterior angles of a polygon is equal to
Regular Polygon: All sides and angles are congruent
Slope intercept form of a line: y = mx+b ; m = slope and b = y-intercept
Standard form of a line: Ax + By = C
Parallel Lines: Lines that have the SAME slope
Perpendicular lines: Lines whose slopes are negative reciprocals
Unit 4 – Congruent Triangles
Properties of Congruence
Reflexive Property:
Symmetric Property:
Transitive Property
Third Angle Theorem: If two angles in a Triangle are congruent, then the third angle is also congruent
SSS – Two triangles are congruent if their sides are congruent
SAS – Two Triangles are congruent is two consecutive sides and the angle between them are congruent
ASA – Two triangles are congruent if two consecutive angles and the side between them are congruent
AAS – Two triangles are congruent if two consecutive angles and the third side are congruent
CPCTC – Corresponding parts of congruent triangles are congruent
HL – In a right triangle if the hypotenuse and legs are congruent, then the triangles are congruent
Isosceles Triangle theorem:
If two sides of a triangle are congruent, the base angles are congruent.
If the base angles are congruent the sides are congruent.
Unit 5 –Relationships within Triangles
Midsegment: A midsegment is a segment connecting the midpoints of two of the sides
Theorem: The midsegment is half the length of the third side and parallel to it.
Bisector: A line that cuts the segment (or angle) into two equal parts.
Concurrent: Three or more lines that intersect at the same point
Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle
Incenter: The point of Concurrency of the
angle bisectors of a triangle
Medians: The medians of a triangle are
concurrent at a point that is 2/3 the distance
from the vertex to the midpoint on the
opposite side. The median is drawn by
drawing a line from the vertex of an angle to
the midpoint on the other side.
Centriod: The point of concurrency
connecting the medians of a triangle
Altitudes: A line segment connecting a point
on the vertex angle to the side at a right angle
Orthocenter: The point of concurrency of the medians of a triangle
Triangle Properties:
1) The largest angle is opposite the longest side, the smallest angle is opposite the shortest side.
2) The sum of any two sides must be greater than the third side
Unit 6 - Quadrilaterals
Special Quadrilaterals
Parallelogram: Opposite sides are parallel and congruent
Rhombus: Four congruent sides
Rectangle: Four right angles
Square: Four right angles and four equal sides
Kite: Two pairs of adjacent sides congruent, and no opposite sides congruent
Trapezoid: Only one pair of parallel sides
Parallelogram properties(and methods of proof)
1) Opposite sides are congruent
2) Opposite sides are parallel
3) Opposite angles are congruent
4) Diagonals Bisect each other
P1) If both pairs of sides are congruent, then it is a parallelogram
P2) If both pairs of opposite angles are congruent, then it is a parallelogram
P3) If the diagonals of a parallelogram bisect each other, then it is a parallelogram
P4) If one pair of opposite sides are BOTH parallel and congruent, then it is a parallelogram
Special Parallelogram Properties(and methods of proof)
1) Each diagonal of a rhombus bisects the angles of a rhombus
2) The diagonals of a Rhombus are perpendicular
3) The diagonals of a rectangle are congruent
P1) If one diagonal of a parallelogram bisects two angles, then it is a rhombus
P2) If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
P3) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle
Trapezoids and Kites(and methods of proof)
1) The base angles of an Isosceles trapezoid are congruent
2) The diagonals of an Isosceles triangle are congruent
3) The Diagonals of a Kite are perpendicular
Trapezoid Midsegment Theorem
Unit 7 – Similar Triangles
Similarity – Polygons are similar if there angles are congruent corresponding sides are proportional
AA – Two Triangles are similar if two angles from one triangle are congruent to two angles in the other
SAS Similarity – Two triangles are similar is two sides are in proportion and the angle between is
SSS Similarity – If three sides are in proportion then the triangles are similar
Geometric Mean –
Side Splitter Theorem – If a line is parallel to one side of a triangle and intersects two other sides, then it
divides those sides proportionally.
Pythagorean Theorem – a2 + b2 = c2 in a right triangle where “c” is the hypotenuse. And the it is true…If
a2+b2 = c2 then it is a right triangle.
Common Pythagorean Triples:
Special Right Triangles –
Sin(x) = Opposite
Cos(x) = Adjacent
Tan(x) = Opposite
Unit 8 - Circles
Standard form of a Circle (x-a)2 + (y-b)2 = r2
Arc Properties –
Area of a Circle =
Tangent Lines to a Circle –
A tangent line is always perpendicular to the radius of the circle.
Chord Properties –
1) Congruent central angles have congruent chords
2) Congruent chords have congruent arcs
3) Congruent Arcs have congruent central angles
4) Chords equidistant from the center are congruent
5) A diameter that is perpendicular to the chord, bisects the chord(diagram)
Unit 9 - Transformations
Transformation – A geometric Figure that can change size, shape or position
Preimage – The original figure
Image – the resulting figure after a transformation
Isometry – a transformation that keeps the figures congruent(Only a dilation will change Isometry)
Composition – A combination of two or more transformations
Reflection – a flip (opposite orientations)
Reflection over the x-axis  (a,b) = (a,-b)
Reflection over the y-axis  (a,b) = (-a,b)
Rotation – a turn
900 around the origin (a,b) = (-b,a)
1800 around the origin  (a,b) = (-a,-b)
Dilation – increase or decrease in size of figure
Translation – A movement of a figure
Symmetry – an Isometry that maps a figure onto itself(line, rotational, point)
Unit 10 - Solids
Lateral Area = Sum of the areas of the lateral faces (do not include the bases)
Surface area = sum of the areas of the faces (include the bases)
Lateral area of a cone =
Volume of a Prism = Area of the base x height
Volume of a pyramid = 1/3 x area of the base x height
Surface area of a sphere =
Volume of a sphere = 4/3
Area and volumes of Similar solids:
If two figures are in the ratio of A:B
The areas are in the ration of A2:B2
The volumes are in the ration of A3:B3
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