Geometry Review
Logic:
Truth Tables:
p
q
p^q
p
q
pνq
p
q
p→q
p
q
p↔q
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F
T
F
T
T
F
F
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
F
F
F
F
F
F
F
T
F
F
T
Laws:
1. Law of Detachment
2. Law of the Contrapositive
p→q
p→q
p
∴~q → ~p
∴q
3. Law of Modus Tollens
4. Chain Rule
p→q
p→q
~q
q→r
∴ ~p
∴p→r
5. Law of Disjunctive Inference
6. Law of Double Negation
pνq
pνq
~ (~p)
~p
~q
∴p
∴q
∴p
7. De Morgan’s Laws
8. Law of Simplification
~ (p ^ q)
~ (p ν q)
p^q
p^q
∴ ~p ν ~q
∴ ~p ^ ~q
∴p
∴q
9. Law of Conjunction
p
10. Law of Disjunctive Addition
p
∴pνq
q
∴p^q
Quantifiers:
Universal Quantifier:
∀𝒙 : all, for all, every, for every
Existential Quantifier
∃𝒙 : there exists, at least one, for some
The negation of “∀𝑥 p” is “∃𝑥 ~p”
The negation of “∃𝑥 p” is “∀𝑥 ~p”
Undefined Terms:
Point: A dot, a location, represented with capital letters
Line: Stringing infinite amounts of points together
Plane: When you string together lines
Defined Terms (converse must work):
Line Segment: If it is a line segment then it is a piece/part of a line with two endpoints
Ray: If it is a ray, then it is a part or a piece of a line with one endpoint
Angle: If it is an angle, then it is the union of two rays with the same endpoint.
Acute Angle: If an angle is an acute angle, then it is less than 90°
Right Angle: If it is a right angle, then it is 90
Obtuse Angle: If an angle is an obtuse angle, then it is more than 90° but less than 180
Straight Angle: If straight angle then it is the union of two opposite rays with the same endpoint
Triangle: If it is a triangle, then it is the union of three noncollinear points’ segments
Midpoint: If it is a midpoint, then it divides a segment into two congruent parts
Bisector: If it is a bisector, then it divides an angle or segment into two congruent parts
Trisector: If it is a trisector, then it divides a segment of angle into 3 congruent parts
Perpendicular: If 2 lines are perpendicular, then they intersect to form a right angle
Complementary: If the sum of 2 angles is 90 , then they are complementary
Supplementary: If the sum of 2 angles is 180 , then they are supplementary
Congruent Triangles: If 2 triangles are
, then their corresponding sides and angles are
Circle: If it is a circle, then it is the set of all points that are equidistant from one point
Median: If it is a median, then it connects the vertex of a triangle to the opposite sides midpoint
Altitude: If it is the altitude, then it is a segment that goes through the vertex of a triangle and is
Regular Polygon: If it is a regular polygon then all sides and angles are ≅
Definitions:
Intersection: overlap at one point
Union: everything together
Vertex: the common endpoint of two rays
Congruent: two segments or angles that have the same measure
Collinear: points on the same line (two points are always collinear)
Noncollinear: points on different lines
Opposite Rays: rays with a common endpoint going in opposite directions (form a line)
Adjacent: two angles with the same vertex and share a ray (next to each other)
Probability: # of winners/ # of possibilities
Interchangeable: 2 angles or segments with the same measure which can be substituted for each other
Vertical Angles: angles that are opposite to each other when two lines intersect
Corresponding: in the same spot
Center: the one point that all other points are equidistant from
Radius: the segment connecting the center point to another point on the circle
Auxiliary Lines: a line or a segment that you add to a diagram
Diameter: two radii’s
Coplanar: same plane
Non-Coplanar: different planes
Parallel Lines: coplanar lines that never intersect
Skew Lines: non-coplanar lines that never intersect
Transversal: a line that intersects two or more lines
Interior: the angles between the two lines that are being intersected
Exterior: the angles outside the two lines being intersected
Alternate: angles on different sides of the transversal
Consecutive: angles on the same side of the transversal
Alternate Interior: angles on different sides of the transversal, with different vertices and they
are btwn the two lines being intersected
Alternate Exterior: angles on different sides of the transversal, with different vertices, and they
are outside the two lines being intersected
Consecutive Interior: angles on the same side of the transversal, with different vertices, and
they are btwn the two lines being intersected
Consecutive Exterior: angles on the same side of the transversal, with different vertices, and
they are outside the two lines being intersected
Corresponding Angles: angles in the same place
Exterior Angle: the angle formed by extending a side of a triangle (each vertex has two)
Midsegment: the segment connecting two midpoints on a triangle
Theorems (converse doesn’t have to work):
1. If right angles, then
2. If straight angles, then
3. If 2 angles are complementary/supplementary to the same angle, then they are
4. If 2 angles are complementary/supplementary to
angles, then they are
5. Addition Property:
≅ parts + ≅ parts = ≅ wholes
≅ parts + same = ≅ wholes
6. Subtraction Property:
≅ wholes – ≅ parts = ≅ parts
≅ wholes – same = ≅ parts
7. Multiplication Property:
parts divided midpoint/bisector/trisector
wholes
8. Division Property:
wholes divided midpoint/bisector/trisector
9. Transitive Property:
If 2 parts are
to same parts, then they are
If 2 parts are
to
parts, then they are
parts
10. Substitution Property:
If 2 segments/angles are
, then they can be substituted for each other
11. If 2 angles are vertical angles, then they are
12. If 2 angles are supplementary and
to each other, then they are right angles
13. Triangle Inequality Theorem
The sum of any two sides of a triangle is greater than the length of the third side, x + y > z
14. CPCTC
Corresponding Parts of Congruent Triangles are Congruent
15. If it is the radii of the same or
16. If
circles, then it is
then
17. The Equidistance Theorems:
a. E1 if two points are equidistance from the endpoints of a segment, then those
points form the
bisector of the segment
b. E2 if a point is on the
bisector of a segment, then that point is equidistance
from the endpoints of the segment
18. Lies on or Passes Through
Converse works too to prove parallel lines
If you know the
bisector of a segment, and a point equidistant from the endpoint of
that segment, then the point lies on the
bisector
19. If parallel lines cut by transversal
alternate interior angles
20. If parallel lines cut by transversal
alternate exterior angles
21. If parallel lines cut by transversal
corresponding angles
22. If parallel lines cut by transversal
consecutive interior angles supplementary
23. If parallel lines cut by transversal
consecutive exterior angles supplementary
24. If two lines are
to a third line, then the lines are parallel
25. Exterior Angle Theorem
The measure of the exterior angle of a triangle is = to the sum of the interior angles
furthest away from it
26. Midline Theorem
The midsegment of a triangle is parallel to the third side
The midsegment of a triangle is half the length of the third side
27. If it is the median to the base of an isosceles
, then it is also the altitude
28. No Choice Theorem
If two angles of one triangle are ≅ to two angles of another triangle the third angles are ≅
29. AAS
If two corresponding angles are
then the
and a corresponding side is
‘s are
30. Sum of Interior Angles = 180 (n-2)
31. Sum of Exterior Angles = 360°
32. Diagonals =
Postulates:
1. SSS:
If 3 pairs of corresponding sides are
, then
‘s
2. SAS:
If 2 corresponding sides are
, and the angle btwn them is
, then
‘s
3. ASA:
If 2 corresponding angles are
and the segment btwn them is
, then
‘s
4. HL:
If two right triangles,
hypotenuses,
legs, then they are
‘s
5. Parallel Postulate
Through any point in the world, you must be able to draw a line parallel to another line
Types of Triangles:
Angle
Sides
1) Right Triangle
1 right angle, 2 acute angles
leg
Hypotenuse
leg
2) Acute Triangle
3 acute angles
3) Obtuse Triangle
1 obtuse angle, 2 acute angles
1) Equilateral Triangle
All 3 sides are
2) Scalene Triangle
All 3 sides are different
3) Isosceles Triangle
At least 2 sides are
leg
4) Equiangular Triangle
All 3 angles are
leg
base
Assumptions:
Can Assume:
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Lines
Collinearity
Relative location of points
Intersection
Straight lines/angles
Supplementary
Vertical Angles
Corresponding
Radii
Coplanar
Non-Coplanar
Transversal
Alternate Interior
Alternate Exterior
Consecutive Interior
Consecutive Exterior
Exterior Angles
Can’t Assume:















Right Angles
Congruency
Measurements
Midpoints
Bisectors
Trisectors
Perpendicular
Complementary
Circles
Triangles
Median
Altitude
Parallel Lines
Midsegments
Quadrilaterals
Properties of Quadrilaterals
Parallelograms:
D 1. If parallelogram
both pairs of opposite sides are parallel
T 2. If parallelogram
opposite sides are
T 3. If parallelogram
opposite angles are
T 4. If parallelogram
consecutive angles are supplementary
T 5. If parallelogram
the diagonals bisect each other
Rectangles:
D 1. If rectangle
it is a parallelogram with at least one right angle
T 2. If rectangle
all angles are right angles
T 3. If rectangle
the diagonals are
Rhombuses:
D 1. If rhombus
it is a parallelogram with a pair of consecutive sides
T 2. If rhombus
all sides are
T 3. If rhombus
the diagonals are
T 4. If rhombus
the diagonals bisect the angles
bisectors of each other
Squares:
D 1. If square
it is a parallelogram, rectangle, and rhombus
T 2. If square
the diagonals form four isosceles right triangles
Kites:
D 1. If kite
two distinct pairs of consecutive sides are
T 2. If kite
the obtuse angles are
T 3. If kite
the longer diagonal is the
T 4. If kite
the longer diagonal bisects the angles
bisectors of the shorter
Isosceles Trapezoid:
D 1. If isosceles trapezoid
the nonparallel sides are
T 2. If isosceles trapezoid
the upper and lower base angles are supp
T 3. If isosceles trapezoid
upper/lower base angles are
T 4. If isosceles trapezoid
diagonals are
How to Prove Quadrilaterals:
Parallelogram:
1. If both pairs of opposite sides parallel
parallelogram
2. If both pairs of opposite sides ≅
parallelogram
3. If both pairs of opposite angles ≅
parallelogram
4. If consecutive angles supplementary
5. If diagonals bisect each other
parallelogram
parallelogram
6. If one pair of opposite sides are parallel and ≅
parallelogram
Rectangle:
1. If it is a parallelogram with one right angle
2. If it is a parallelogram with ≅ diagonals
rectangle
rectangle
Rhombus:
1. If it is a parallelogram with a pair of consecutive sides ≅
rhombus
2. If it is a parallelogram and either diagonals bisects two angles
3. If the diagonals are
bisectors of each other
rhombus
rhombus
Square:
1. If it is a rectangle, and rhombus
square
Isosceles Trapezoid:
1. If it is a trapezoid and the nonparallel sides are ≅
isosceles trapezoid
2. If it is a trapezoid and the upper/lower base angles are ≅
3. If it is a trapezoid and the diagonals are ≅
isosceles trapezoid
isosceles trapezoid