Geometry Proofs Flashcards
Three types of proofs
Flow proof, Paragraph proof, Two Column Proof
What are postulates 2.1 through 2.15 about?
Basic rules about points and lines, and other rules about segments and angles
Postulates 2.1-2.7
-if there are two points there is exactly one line going through them
-if there are 3 non collinear points there is exactly one plane going through them
-Every line has at least 2 points that define it
-Every plane has at least three points that define it
-If there are two points on a plane, the line containing them is also on that plane
-Two lines intersect ant one point
-Two planes intersect at one line
Ruler postulate (2.8)
The distance between two points on a line can be measured
-Used rarely
Segment Addition Postulate (2.9)
If A, B, and C are collinear and B is between A and C then AB+ BC= AC
-Used to simplify connecting segments into one segment
Protractor Postulate (2.10)
Angles can be measures
- Used Rarely
Angle Addition Postulate (2.11)
point D is on the interior of angle ABC if and only if ABD + DBC = ABC
-Used to simplify adjacent angles and split them apart
Corresponding Angles Postulate
If two parallel Lines are cut by a transversal then each pair of corresponding angles are congruent.
-Use to find angle measures
Converse of Corresponding Angles Postulate
If the pairs of corresponding angles are congruent, the the lines being cut by a transversal are parallel.
-Used to prove parallel lines
Parallel Postulate
If Given a line and a non collinear point, the point has exactly one line running through it that would be parallel to the given line
-Rarely Used
Perpendicular Postulate
If Given a line and a non collinear point, the point has exactly one line running through it that would be parallel to the given line
-Rarely Used
What are the properties of real numbers
-addition, subtraction, multiplication, division
- reflexive (a=a), symmetric (a=b, b=a), transitive (a=b, b=c, then a=c), substitution, distributive (a(b+c)=ab+ac)
Midpoint theorem
2.1 M is the midpoint of AB if and only if AM=MB
-use to convert _ is the mid point of _ into an equation
-often first step
2.2 Properties of segment congruence
Reflexive, Symmetric, Transitive
-not commonly used
2.3 Supplement Theorem, and 2.4 Complement theorem
- If there is a linear pair, the two angles are supplementary
-If the non common sides of adjacent angles form a right angle, the angles are complementary
-Used to convert image/given to equation
2.5 Properties of segment congruence theorem
Reflexive Symmetric, Transitive
-not commonly used
2.6 Congruent Supplements, and 2.7 Congruent Complements (THEOREMS)
If two angles complement/supplement to the same angle then they are are congruent
-used from drawings
2.8 Vertical Angle theorem
Vertical Angles are congruent
- used from drawings
2.9 through 2.13, right angle theorems
- Perpendicular lines intersect to form 4 right angles
- All right angles are congruent
- perpendicular lines form congruent adjacent angles
- if two angles are congruent and supplementary they are right
- If two angles are congruent and form a linear pair they are right
2.14 Alternate Interior Angles theorem and 2.20 Alternate Interior Angles Converse theorem
-AIA are always congruent if transversal cuts parallel lines
-If AIA are congruent then transversal cuts parallel lines
-used to prove angle measure or that lines are parallel
2.17 CIA theorem, and 2.21 converse theorem
-CIA are always congruent if transversal cuts parallel lines
-If CIA are congruent then transversal cuts parallel lines
-used to prove angle measure or that lines are parallel
2.16 AEA Theorem, and 2.22 AEA converse theorem
-AEA are always congruent if transversal cuts parallel lines
-If AEA are congruent then transversal cuts parallel lines
-used to prove angle measure or that lines are parallel
2.17 Perpendicular transversal theorem
If a transversal is perpendicular to one of two parallel lines it is perpendicular to the other
-used to prove angle measure=90
2.18 slope of parallel lines, and 2.19 slope of perpendicular lines
-parallel lines have the same slope
-perpendicular lines have opposite slope
2.24 Two Lines Equidistant from a Third Theorem
If two lines are equidistant from a third they are parallel
Definition of congruence
Used Often! switch from congruence to equals and back
4.1 Triangle Sum theorem
The sum of all the angles in a triangle is 180
4.2 Exterior angle theorem
The exterior angle is the sum of the two remote interior angles
4.3 Third angles theorem
If two of the angles in a triangle are congruent to the angles in another triangle, the third angles are congruent too
4.4 Triangles properties of congruence theorem
reflexive, symmetric, transitive
Definition of congruent polygons
Used like definition of congruence but for polygons
AAS, SSS, SAS, ASA (THEOREMS/POSTULATES?)
A= angle S= side
If two triangles have AAS, SSS, SAS, or ASA congruent then the triangles are congruent
CPCTC
stands for, corresponding parts of congruent triangles are congruent.
- use to prove corresponding parts of triangles are equal in measure
What do L, A, S, and H stand for?
L- leg (right triangle)
H- hypotenuse (right triangle
A- angle
S- side
What are the right triangle congruence theorems?
LL, LA, HL, AH
right triangles are congruent if any of these parts are given congruent.
What are the properties of an equilateral triangle
The legs are congruent, the angle not opposite to either leg is the vertex, the angles that are opposite to the legs are the base angles
4.10 Isosceles triangle theorem
The base angles in an isosceles triangles are congruent
4.3 properties of an equilateral triangle
a triangle is equilateral if an only if its equal-angular
4.4 properties of an equilateral triangle
each angle of an equilateral triangle measures 60
