Geometry
Interior formula
(n-2) * 180 I = -------------- 2
The sum of all three angles in a triangle equal ?
180 degrees
Exterior formula
360 E = -------- n
The acute angle of a right triangle are ?
Complementary (The sum of two angles equal 90 degrees)
If two lines are parallel to a third line then these lines are
Parallel to each other
collinear
point on the same line
Strategy for indirect proof
1. Assume temporarily that (the opposite of the prove statement) 2. Then.....try to get a statement that contradict the given 3. But this contradict the given fact that (given) 4. Therefore, the temporarily assumption that......#3......is false 5. It follows that (the prove statement "GIVEN")............
Write the indirect proof GIVEN - AB > BC PROVE - B is not the midpoint of SEGMENT AC
1. Assume temporarily that B is the midpoint of segment AC 2. Then AB is not > BC 3. But this contradict the given fact that AB is > BC 4. Therefore, the temporarily assumption that B is midpoint of segment AC is false 5. It follows that B is not the midpoint of AC
How to construct congruent angle
1. Put needle on vertex n draw circle 2. Put needle on outside of circle but extend bigger than vertex, do this both side to get line
Properties of congruent triangles
1. Reflexives property of congruence: /\ABC congruent to /\ABC 2. Symmetric property of congruence: /\ABC congruent to /\DEF, then /\DEF congruent to /\ABC 3. Transitive property of congruence: if /\ABC congruent to /\DEF and /\DEF congruent to /\GHI then /\ABC is congruent to /\GHI
Five ways of proving congruent triangles
1. SSS (side side side - if 3 side of one triangle are congruent to the 3 side of second triangle, then the triangle are congruent ) 2. SAS (side angle side - if two sides and the included angle of one triangle are congruent to two side and the included angle of a second triangle the the triangle are congruent) 3. ASA (angle side angle - if two angle and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the second triangle is congruent ) 4. AAS ( angle angle side - if two angles and a non-included side of a triangle are congruent to two angles and a non-included sides of a second triangle, then the triangles are congruent) 5. HL (Hypotenuse-Leg - when a leg and a hypotenuse of one triangle are congruent to a leg and a hypotenuse of another right triangle then the triangle are congruent)
Name, in order, the five parts of the formal proof of a theorem.
1. statement 2. drawing 3. given 4. prove 5. proof
Each angle of an equiangular triangles measures ?
60 degrees
Polygon
A closed plane figure whose sides are lines segments that intersect only at the endpoints
Paralellogram
A quadrilateral in which both opposite sides are parallel
Triangles classified by angles
Acute - all angles acute measure between 0 and 90 degrees Obtuse - one obtuse angle measure more than 90 degrees Right - one right angle Equiangular - all angles congruent
give the meaning of CD, CD ^ ----, CD, and CD ^-->
CD^ means line CD CD^ --- means line segment CD CD means the measure or Length of CD^ ---
Indirect proof
Conditional (or implication) - if P then Q Converse of conditional - if Q then P Inverse of condition - if not P then Q Contrapositive of condition - if not Q then not P
If two angles of a triangle are congruent to two angles of another triangle then the third angles are also !
Congruent
If two lines are cut by a transversal then the alternate interior angles are ?
Congruent
If two parallel line are cut a transversal then the alternate exterior angles are ?
Congruent
Two angle are ? If one coincides (fit perfectly over) the other
Congruent
CPCTC
Corresponding parts of congruent triangles are congruent
Transitive property
It keep going..if A=B and B=C and C=D that mean A=D so the first one always equal the last and so does everything in between.
Reflexive property
Mirror or reflection...when A is congruent to itself
If two lines are cut by a transversal so that the alternate exterior angles are congruent then these line are
Parallel
If two lines are cut by a transversal so that the corresponding angles are congruent then these lines are
Parallel
If two lines are cut by a transversal so that the exterior angle on the same side of the transversal are supplementary then these line are
Parallel
If two lines are cut by a transversal so that the same side interior angles are supplementary, then these line are
Parallel
If two coplanar lines are each perpendicular to a third lines, then these lines are
Parallel to each other
A concave polygon have at least one ?
Reflex angle
Sum of the measures
S = (n-2) * 180
Triangles classified by congruent sides
Scalene - none Isosceles - two sides Equilateral - 3 sides
If two parallel lines are cut by a transversal then the interior angles on the same side of the transversal are
Supplementary (the sum of measure of two angles equal 180 degrees)
Symmetric property
Symmetry, same on both side, its symmetrical
Convex polygon
The angles measure are between 0 and 90 degrees
A diagonal of a polygon is a line segment that join ?
Two nonconsecutive vertices
lines
a line is an infinite set of points
transversal
a line that intersect 2 or more lines at distinct points
segment
a segment is a line that could contain line segment as part of it *a straight line*
statement
a set of words and symbols that collectively make a claim that can be classified as true or false
postulate
a statement that is assumed to be true
intuition (based on how you feel)
a sudden insight allows one to make a statement without applying any formal reasoning
bisected angle
an angle divided into 2 congruent angles
obtuse angle
an angle that measure between 90 and 180 degrees
corresponding angles
angles that that lie in the same relative positions (above the parallel line and left of the transversal)
adjacent angles
angles who have a common vertex and a common side between them
if two line intersect, they intersect where?
at a point
circle
collection of point that are equal distance from the center
bisect
cut in half
alternate exterior
exterior (ext.) angles that lie on opposite sides of the transversal
same side exterior angles
exterior angles that lie on the same side of t transversal
conditional (or implication) converse of conditional inverse of conditional contrapositive of conditional
if P then Q if Q, then P if not P, then not Q if not Q, then not P
alternate interior
interior (int.) angle'\s that lie on opposite sides of the transversal
same side interior
interior angles that lie on the same side of the transversal
disjunction
is false only when P and Q are both false ( 4 + 3 = 7 or Cypress College is in the city of Fullerton )
conjunction
is true only when P and Q are both true ( 4 + 3 = 7 and Cypress College is in the city of Cypress )
parallel line
lines that lie in the same plane but do not intersect
straight angle
measure 180 degrees **double right angle**
acute angle
measure between 0 and 90 degrees
right angle
measure exactly 90 degrees
reflex angle
measures more than 180 degrees
Total number of diagonals
n(n-3) D = --------- 2
skew
neither parallel nor at right angles to a specified or implied line; askew; crooked.
the measure or length of a line segment is a
number
perpendicular lines
perpendicular line form to make 2 congruent adjacent angles
point
point are represented by a dot labeled with a single capital letter
midpoint
point in the middle
non coplanar points
points not on the same plane
coplanar points
points on the same plane
Does the relation "is greater than" have a reflexive property (consider real number A)? a symmetric property (consider real numbers A and B)? a transitive property (consider real numbers A, B, and C)?
reflexive # cannot be greater than itself. symmetric # cannot be greater than or less than itself. Only transitive # work because #A can be greater than #B and #B is greater than #C.
Does the relation "is less than" for numbers have a reflexive property (consider one number)? a symmetric property (consider two numbers)? a transitive property (consider three numbers)?
reflexive # cannot be less than itself. symmetric # cannot be less or greater than itself. Only transitive # will work because #A can be less than #B and #B is less than #C.
Does the relation "is complementary to" for angles have a reflexive property (consider one angle)? a symmetric property (consider two angles)? a transitive property (consider three angles)?
reflexive angles cannot be complementary. Symmetric angles can be complementary because two angles can add up to be 90 degrees. Transitive angle cannot be complementary because complementary is the combination of 2 angles not three.
Does the relation "is a brother of" have a reflexive property (consider one male)? a symmetric property (consider two males)? a transitive property (consider three males)?
reflexive brother, one cannot be his own brother. symmetric brother, yes because brother A is a brother to brother B. transitive brothers, Yes because brother A is a brother of Brother B and brother B is a brother of brother C therefore brother A and brother C are also brother.
Does the relation "is perpendicular to" have a reflexive property (consider line L)? a symmetric property (consider lines L and M)? a transitive property (consider lines L, M, and N)?
reflexive line cannot be perpendicular to itself. symmetric line can be perpendicular. transitive line cannot be perpendicular because there's no way to make 3 line perpendicular to each other.
Venn Diagrams
represent the law of detachment
bisector
separating into 2 congruent parts
If two parallel lines are cut by a transversal then the exterior angles on the same side of the transversal are ?
supplementary
If two parallel lines are cut by a transversal, then the interior angles side of the transversal are ?
supplementary
deduction (a law like 1+1=2)
the knowledge and acceptance of selected assumptions guarantee the truth of a particular conclusion
negation *also called logical complement*
the negation of a given statement P, denoted by ~P (which is read "NOT P"), makes a claim opposite that of the original statement. If the given statement is true, its negation is false, and vice versa
vertical angles
the nonadjacent angles formed by intersecting lines
Two triangles are congruent if the six parts of the first triangles is congruent to what?
to the six corresponding parts of the second triangles Note: the reverse is the same
a plane is ________ dimensional
two
congruent angles
two angles that have the same measure
supplementary angles
two angles whose sum is 180 degrees
complementary angles
two angles whose sum is 90 degrees
congruents angles
two angles with equal measure
opposite ray
two rays that share a common endpoint
parallel line
two straight line that never intersect
angle
union of two rays that share an endpoint
induction (if u see something that happen a lot u assumes that it will happen again)
using specific observation and experiments to draw a general conclusion